Device and method for wave-field reconstruction

ABSTRACT

Computing device, computer instructions and method process input seismic data d recorded in a first domain by seismic receivers that travel in water, the input seismic data d including pressure and particle motion measurements, including up-going and down-going wave-fields. A model p is generated in a second domain by solving an inverse problem for the input seismic data d, wherein applying an L transform to the model p describes the input data d. An L′ transform, which is different from the L transform, is then applied to the model p to obtain an output seismic data in the first domain, the output seismic data having a characteristic imparted by the transform L′. The characteristic is related to pressure wave-fields and/or particle motion wave-fields interpolated at positions in-between the input seismic receivers. An image of the surveyed subsurface is generated based on the output seismic dataset.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation Application of U.S. patentapplication Ser. No. 15/275,818, filed on Sep. 26, 2016, which is aContinuation Application of U.S. patent application Ser. No. 14/678,099,filed on Apr. 3, 2015, now U.S. Pat. No. 9,535,181, which is aContinuation Application of International Application No.PCT/EP2014/058623 filed on Apr. 28, 2014, which claims priority andbenefit from U.S. Provisional Application 61/817,193, filed Apr. 29,2013, titled, “Deghost, Redatum, and Interpolation using Multi-componentStreamer Data,” and authored by G. Poole; U.S. Provisional Application61/824,040, filed May 16, 2013, titled, “Deghost, Redatum, andInterpolation using Multi-component Streamer Data,” and authored by G.Poole; U.S. Provisional Application 61/824,521, filed May 17, 2013,titled, “Deghost, Redatum, and Interpolation using Multi-componentStreamer Data,” and authored by G. Poole; U.S. Provisional Application61/911,574, filed Dec. 4, 2013, titled, “Deghost, Redatum, andInterpolation using Multi-component Streamer Data,” and authored by G.Poole; and U.S. Provisional Application 61/931,196, filed Jan. 24, 2014,titled, “Deghost, Redatum, and Interpolation using Multi-componentStreamer Data,” and authored by G. Poole, the entire contents of whichare incorporated herein by reference.

BACKGROUND Technical Field

Embodiments of the subject matter disclosed herein generally relate tomethods and systems and, more particularly, to mechanisms and techniquesfor reconstructing wave-fields (e.g., deghosting, redatuming, denoising,interpolating, etc.) based on seismic data collected with receiverslocated either on streamers or on or close to the ocean bottom.

Discussion of the Background

Marine seismic data acquisition and processing generate a profile(image) of the geophysical structure (subsurface) under the seafloor.While this profile does not provide an accurate location for oil andgas, it suggests, to those trained in the field, the presence or absenceof oil and/or gas. Thus, providing a high-resolution image of thesubsurface is an ongoing process for the exploration of naturalresources, including, among others, oil and/or gas.

During a seismic gathering process, as shown in FIG. 1, a vessel 110tows plural detectors 112, which are disposed along a cable 114. Cable114 together with its corresponding detectors 112 are sometimes referredto, by those skilled in the art, as a streamer 116. Vessel 110 may towplural streamers 116 at the same time. Streamers may be disposedhorizontally, i.e., lie at a constant depth z₁ relative to the oceansurface 118. Also, plural streamers 116 may form a constant angle (i.e.,the streamers may be slanted) with respect to the ocean surface asdisclosed in U.S. Pat. No. 4,992,992, the entire content of which isincorporated herein by reference.

Still with reference to FIG. 1, vessel 110 may also tow a seismic source120 configured to generate an acoustic wave 122 a. Acoustic wave 122 apropagates downward and penetrates the seafloor 124, eventually beingreflected by a reflecting structure 126 (reflector R). Reflectedacoustic wave 122 b propagates upward and is detected by detector 112.For simplicity, FIG. 1 shows only two paths 122 a corresponding to theacoustic wave. Parts of reflected acoustic wave 122 b (primary) arerecorded by various detectors 112 (recorded signals are called traces)while parts of reflected wave 122 c pass detectors 112 and arrive at thewater surface 118. Since the interface between the water and air is wellapproximated as a quasi-perfect reflector (i.e., the water surface actsas a mirror for acoustic waves), reflected wave 122 c is reflected backtoward detector 112 as shown by wave 122 d in FIG. 1. Wave 122 d isnormally referred to as a ghost wave because it is due to a spuriousreflection. Ghosts are also recorded by detector 112, but with a reversepolarity and a time lag relative to primary wave 122 b if the detectoris a hydrophone. The degenerative effect that ghost arrival has onseismic bandwidth and resolution is known. In essence, interferencebetween primary and ghost arrivals causes notches, or gaps, in thefrequency content recorded by detectors.

Recorded traces may be used to determine the subsurface (i.e., earthstructure below surface 124) and to determine the position and presenceof reflectors 126. However, ghosts disturb the accuracy of the finalimage of the subsurface and, for at least this reason, various methodsexist for removing ghosts, i.e., deghosting, from the acquired seismicdata. These methods were designed for deghosting seismic data recordedwith horizontal or slanted streamers.

The above-discussed methods are not appropriate for seismic datacollected with new streamer configurations, e.g., having a curvedprofile as illustrated in FIG. 2. Deghosting methods for streamershaving a curved profile have recently been developed, mainly by theassignee of this application, as later discussed. Such a configurationhas a streamer 252 with a curved profile. The curved profile may haveany shape. One example of a curved profile is defined by threeparametric quantities, z₀, s₀ and h_(c). Note that not the entirestreamer has to have the curved profile. The first parameter z₀indicates the depth of the first detector 254 a relative to the water'ssurface 258. Second parameter so is related to the slope of the initialpart of streamer 252 relative to a horizontal line 264. The exampleshown in FIG. 2 has initial slope so equal to substantially 3 percent.Other values may be used. Note that the streamer 252 profile in FIG. 2is not drawn to scale because a slope of 3 percent is relatively slight.Third parameter h_(c) indicates a horizontal length (distance along theX axis in FIG. 2 measured from first detector 254 a) of the streamer'scurved portion. This parameter may be in the range of hundreds tothousands of meters.

For such streamers, a deghosting process has been disclosed, forexample, in U.S. Pat. No. 8,456,951 (herein '951) authored by RSoubaras, the entire content of which is incorporated herein. Accordingto the '951 patent, a method for deghosting uses joint deconvolution formigration and mirror migration images to generate a final image of asubsurface. Deghosting is performed at the end of processing (during animaging phase) and not at the beginning, as with traditional methods.Further, the '951 patent discloses that no datuming step is performed onthe data.

Another method that addresses variable-depth data is disclosed by U.S.patent application Ser. No. 13/334,776 (herein '776) authored by G.Poole. This method uses a surface datum tau-p model that representsinput shot data. A transform from the tau-p model to a shot domain(offset-time) combines the operations of redatuming and reghosting. Theuse of variable-depth streamer data combined with reghosting ensuresthat a single point in the tau-p domain satisfies a range of differentghost lags, therefore, making use of variable-depth data notchdiversity, which ensures effective receiver deghosting.

FIG. 3 illustrates how one point 302 in the tau-p domain 320 reversetransforms to a pair of lines 304 and 306 in time-space domain 310, atactual streamer position and mirror streamer datum, respectively. Theenergy relating to mirror cable datum has reverse polarity (−1) comparedto the cable datum (+1) as also illustrated in FIG. 3. Note that thetime-space (or time-offset) domain 310 has time t on the vertical axis,and offset h between the receiver recording the wave and the sourcegenerating the wave, along the X axis, while tau-p domain 320 has thetau coordinate (intercept in the time-space domain) along the Y axis,and the p coordinate (slope in the time-space domain) along the X axis.Application '776 uses a least squares formulation given by:

d=Lp  (1)

or, in the expanded matrix form,

$\begin{matrix}{{\begin{pmatrix}d_{1} \\d_{2} \\d_{N}\end{pmatrix} = {\left( {e^{{- 2}\pi if\tau_{p\; r}} - e^{{- 2}\pi if\tau_{gh}}} \right)\begin{pmatrix}p_{1} \\p_{2} \\p_{3} \\p_{M}\end{pmatrix}}},} & (2)\end{matrix}$

where column vector d contains a frequency slice from the shot domaindata (known), column vector p contains the surface datum tau-p model(unknown), and matrix L makes the transform (known) from the surfacetau-p model to the input shot data. Matrix L combines the operations ofredatuming and reghosting.

The time shifts for primary (up-going) and ghost (down-going) wavefields are given by:

τ_(pr)=(h _(n) +Δh)s _(m)−Δτ  (3)

τ_(gr)=(h _(n) −Δh)s _(m)+Δτ  (4)

where h_(n) is the offset of a given trace in column vector d, s_(m) isthe slowness of a given trace in the surface tau-p model, Δh is theoffset perturbation as described in the '776 application, and Δτ is thetemporal perturbation as also described in the '776 application.Equation (1) can be solved in the time or spectral (e.g., frequency)domain using linear inversion. The method can be applied on the wholeshot (cable-by-cable) or in spatial windows of a user-defined number ofchannels.

However, existing methods relate to pressure measurements made, forexample, by hydrophones. Currently, the new streamer generation iscapable of measuring not only pressure but also particle motion data,e.g., displacement, velocity, differential pressure, acceleration, etc.Thus, there is a need to process not only pressure measurements, butalso particle motion data. Accordingly, it would be desirable to providesystems and methods with such capabilities.

SUMMARY

According to an embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded in a first domain by seismic receivers that travel in water,the input seismic data d including up-going and down-going wave-fields;generating a model p in a second domain to describe the input seismicdata d; and processing with a processor the model p to obtain an outputseismic dataset indicative of the down-going wave-field andsubstantially free of the up-going wave-field.

According to another embodiment, there is method for processing inputseismic data d. The method includes receiving the input seismic data drecorded in a first domain by seismic receivers that travel in water;generating a model p in a second domain, at a datum different from theinput seismic data d, to describe the input seismic data d; andprocessing in a processor the model p to obtain an output seismicdataset indicative of a pressure wave-field.

According to another embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded in a first domain by seismic receivers that travel in water,the input seismic data d including pressure and particle motionmeasurements; generating a model p in a second domain to describe theinput seismic data d, wherein the model p is obtained by solving aninverse problem based on an L transform; and processing, in a processor,with a mathematical transform L′ the model p to obtain, in the firstdomain, an output seismic data having a characteristic imparted by thetransform L′. The mathematical transform L′ is different from themathematical transform L.

According to another embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded in a first domain by seismic receivers that travel in water,wherein the input seismic data includes both pressure and particlemotion measurements; generating a model p in a second domain to describethe input seismic data d; and processing with a processor the model p toseparate multiples and primaries in the second domain, wherein themultiples is multiple energy reflected at the free surface or rockinterface layers in the subsurface.

According to another embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded by seismic receivers that travel in water, the input seismicdata d including particle motion measurements; receiving receiverorientation data that is indicative of seismic receiver orientations;associating the particle motion measurements with the receiverorientation data; and generating with a processor an output seismic databased on the association of the particle motion measurements with thereceiver orientation data. The seismic receiver orientations vary intime and the output seismic data includes a wavefield reconstruction ofthe input dataset.

According to another embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded, in a first domain, by seismic receivers that travel in water,the input seismic data d including pressure and particle motionmeasurements; generating a model p in a second domain to describe theinput seismic data d; and processing in a processor the model p togenerate an output seismic dataset with attenuated noise.

According to another embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded, in a first domain, by seismic receivers that travel in water;generating a model p in a second domain to describe the input seismicdata d; and processing with a processor the model p to generate a outputparticle motion dataset.

According to another embodiment, there is a method for processing inputseismic data d. The method includes receiving the input seismic data drecorded, at a first datum, in a first domain, by seismic receivers thattravel in water, the input seismic data d including up-going anddown-going wave-fields; generating a model p in a second domain todescribe the input seismic data d; and processing with a processor themodel p to obtain an output seismic dataset at a second datum, differentfrom the first datum.

According to still another embodiment, there are computing systems andcomputer-readable mediums including computer executable instructions,wherein the instructions, when executed by a processor, implement one ormore of the methods as noted in the above paragraphs.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate one or more embodiments and,together with the description, explain these embodiments. In thedrawings:

FIG. 1 is a schematic diagram of a conventional seismic data acquisitionsystem having a horizontal streamer;

FIG. 2 is a schematic diagram of a seismic data acquisition systemhaving a curved profile streamer;

FIG. 3 is a schematic diagram of a transform L from a first domain to asecond domain;

FIGS. 4A-C illustrate input multicomponent data and FIG. 4D illustratesoutput deghosted data;

FIG. 5A illustrates an incoming ray and an angle convention associatedwith it;

FIG. 5B schematically illustrates a processing method for obtainingdeghosted and reconstructed data;

FIGS. 6A-C illustrate input multicomponent data and FIG. 6D illustratesoutput deghosted data;

FIGS. 7A-C illustrate input multicomponent data, FIG. 7D illustratesoutput deghosted data, FIG. 7E illustrate various channels and FIG. 7Fillustrate a change in polarity;

FIG. 8 illustrates how a seismic receiver's position changes along astreamer;

FIGS. 9-11 illustrate seismic data redatuming based on inputmulticomponent data;

FIG. 12 illustrates a depth sampling versus slowness for a givenstreamer;

FIG. 13 illustrates inconsistent spatial sampling for output data withredatuming;

FIGS. 14A-G illustrate surface spatial sampling for output seismic data;

FIG. 15 is a flowchart of a method for generating output denoisedpressure data;

FIG. 16 is a flowchart of a method for generating deghosted andinterpolated output seismic data;

FIG. 17 is a flowchart of a method for generating interpolated andobliquity-corrected output seismic data;

FIG. 18 is a schematic diagram of a computing device configured toimplement a deghosting method according to an exemplary embodiment; and

FIGS. 19-26 are flowcharts of methods for processing input seismic data.

DETAILED DESCRIPTION

The following description of the exemplary embodiments refers to theaccompanying drawings. The same reference numbers in different drawingsidentify the same or similar elements. The following detaileddescription does not limit the invention. Instead, the scope of theinvention is defined by the appended claims. The following embodimentsare discussed, for simplicity, with regard to pressure and particlevelocity measurements associated with seismic data. However, theembodiments to be discussed next are not limited to these measurements.Other measurements, e.g., particle displacement and/or particleacceleration measurements, may be used instead of or in addition toparticle velocity measurements. Thus, a generic name for velocity,displacement, pressure gradient, and acceleration measurements isparticle motion data.

Reference throughout the specification to “one embodiment” or “anembodiment” means that a particular feature, structure or characteristicdescribed in connection with an embodiment is included in at least oneembodiment of the subject matter disclosed. Thus, the appearance of thephrases “in one embodiment” or “in an embodiment” in various placesthroughout the specification is not necessarily referring to the sameembodiment. Further, the particular features, structures orcharacteristics may be combined in any suitable manner in one or moreembodiments.

According to an embodiment, hydrophone and vertical particle velocityreceiver data collected from receivers (located on streamer, oceanbottom cable, autonomous vehicles, etc.) are processed as now discussed.

When vertical particle velocity data is available, and assuming 2Dpropagation, the transfer function or operator L (see FIG. 3 andequation (1), i.e., d=Lp) can be extended so that the tau-p model “p”simultaneously satisfies hydrophone and vertical particle velocity data“d” after calibration, i.e.,

$\begin{matrix}{{\begin{pmatrix}h_{1} \\h_{2} \\h_{N_{h}} \\{vz}_{1} \\{vz}_{2} \\{vz}_{N_{p}}\end{pmatrix} = {\begin{pmatrix}{e^{{- 2}\; \pi \; {if}\; \tau_{u}} - e^{{- 2}\; \pi \; {if}\; \tau_{d}}} \\{\cos \; {\theta_{m}\left( {e^{{- 2}\; \pi \; {if}\; \tau_{u}} + e^{{- 2}\; \pi \; {if}\; \tau_{d}}} \right)}}\end{pmatrix}\; \begin{pmatrix}p_{1} \\p_{2} \\p_{3} \\p_{M}\end{pmatrix}}},} & (5)\end{matrix}$

where τ_(u) is the time shift for the up-going wave and τ_(d) is thetime shift for the down-going wave. The time shifts may be definedsimilar to those of equations (3) and (4) or in a different manner,depending on the model used to describe the wave propagation. The top ofthe L matrix and the top of the data vector d relate to the N_(h)hydrophone measurements, and the bottom of the matrix L and the bottomof the data vector d relate to the N_(p) vertical particle velocitymeasurements. The term cos θ_(m) in matrix L represents an obliquityfactor for a given slowness, e.g., sin θ_(m)=±v_(w)|s_(m)|, where s_(m)is the slowness for the m^(th) trace (s/m), and v_(w) is the watervelocity between the receivers and sea surface (m/s). The sign should betaken from the sign of s_(m). The water velocity may be fixed or allowedto vary during the duration of an acquisition. The obliquity factortakes into consideration the inclination θ_(m) of each recorded rayrelative to a receiver orientation and the receiver orientation may bedefined as an angle relative to a vertical axis, e.g., gravity. In otherwords, the receivers in the streamers can have any orientation due totwisting of the streamer. The raw measurements are then rotated to, forexample, a vertical direction and a direction perpendicular to thestreamer. Other directions may be used. Therefore, the obliquity termcorrects recorded amplitudes based on the difference in the orientationof the recording receiver and a direction of an incoming wavefield. Notethat the receiver orientation is different from obliquity. Whileobliquity is an amplitude term only, the receiver orientation has to dowith the orientation of the receivers. In this regard, imagine that thereceiver is a single axis accelerometer. Ideally, the receiver'sorientation may be aligned with the vertical direction (z axis) or thecross-line direction (y axis) of the nominal shooting direction or zerofeather direction (other directions may be preferred, normally twoorthogonal directions). However, the streamer experiences twisting andother movements (e.g., feathering) while being towed in water. Thus, theactual orientation of the receiver is most likely neither along thevertical direction nor along the cross-line direction. The rawmeasurements from the plural receivers may or may not have been rotatedin a plane perpendicular to the streamer at the location of the receiverprior to being received for processing. Further complications areintroduced when the streamer is slanted or has a curved profile becausethe vertical orientation of the receivers is not vertical. A method forcorrectly handling input data that is not ideally aligned with the y andz directions is discussed in the paragraphs related to equations (22) to(28).

It is known that receiver ghost troughs in hydrophone data correspond topeaks on vertical particle velocity receiver data. In the formulationrepresented by equation (5), a single surface datum tau-p model isderived to satisfy both hydrophone and vertical particle velocityreceiver data. As peaks in vertical particle velocity receiver datacorrespond to troughs in hydrophone data, the dual modelling approach ofequation (5) can be more robust than working with either hydrophone orvertical particle velocity measurements in isolation. Once the surfacedatum tau-p model of up-going energy has been found, it may be used tooutput up-going or down-going energy at any datum and offset. Thisresult leads to several options for the 2-D case, a selection of whichis given below:

-   -   1) Output up-going (primary) data at the streamer datum. The        up-going may be subsequently subtracted from original pressure        data to leave down-going (ghost) energy only.    -   2) Output down-going data at the streamer datum. The down-going        may be subsequently subtracted from original pressure data to        leave up-going energy only.    -   3) Output up-going or down-going data at sea surface datum.    -   4) Output up-going or down-going data at a new horizontal datum.        This allows wavefield separated data to be provided at the start        of the processing sequence, so that the data can be processed        with conventional algorithms.    -   5) Output up-going or down-going data at a new variable depth        datum.    -   6) Output up-going and down-going data at a new datum. For        example, this can be useful for time-lapse studies.    -   7) It is also possible to apply deghosting and redatuming in two        independent steps. First deghost the data (leaving it at the        original depths), then redatum it as a separate operation.    -   8) Instead of outputting up-going and down-going wave-fields, it        is possible to output particle motion data at any x-z location        with any orientation. This allows the output of particle motion        data on or between the streamers that is substantially free of        down-going (ghost) or up-going (primary) energy. The particle        motion data may or may not be compensated for obliquity.        The above options can be used to output data at the same offsets        as input data, or at new offsets. These are only a set of        options, and those skilled in the art could easily imagine other        options based on the above-discussed concepts.

One of the above-discussed options has been implemented for exemplarypurposes using equation (5) and is illustrated in FIGS. 4A-D. Inputhydrophone and input vertical particle velocity data (i.e., recordeddata d) is shown in FIGS. 4A and 4B, and the output up-going anddown-going wave-fields are shown in FIGS. 4C and 4D. To obtain theoutput down-going wave-fields (i.e., the ghost), a modified L′ transformis applied to model p, where, for example, the modified L′ transform maybe:

$L^{\prime} = {\begin{pmatrix}e^{{- 2}\; \pi \; {if}\; \tau_{u}} \\{\cos \; {\theta_{m}\left( e^{{- 2}\; \pi \; {if}\; \tau_{u}} \right)}}\end{pmatrix}\; {\left( {6 - a} \right).}}$

Note that modified L′ transform (which in this case is determined bysetting the ghost terms to be zero), when applied to model p, generatesup-going wave-fields d′ in the time-space domain, and this data d′ maybe subtracted from the original seismic data d to obtain down-going (orsurface ghost) wave-fields. If the up-going fields are wanted, themodified L′ transform may have the following form:

${L^{\prime} = {\begin{pmatrix}{- e^{{- 2}\; \pi \; {if}\; \tau_{d}}} \\{\cos \; {\theta_{m}\left( e^{{- 2}\; \pi \; {if}\; \tau_{d}} \right)}}\end{pmatrix}\; \left( {6 - b} \right)}},$

to generate down-going energy which may be subtracted from hydrophonedata. Due to preferential signal-to-noise levels, wave-field separationis often performed by subtraction of an up-going or a down-going datasetfrom the pressure recording. In this case, only the hydrophone terms inthe above equation would be required. Other forms for the modified L′matrix may be used, depending on the desired output data, e.g., newoutput positions x′ and z′ may be selected to reconstruct wave-fields atother positions than the original data. The term “wave-fieldreconstruction” is understood herein to include not only wavefieldseparation, but also re-datuming and/or interpolation or any otheroperation that changes the seismic data's spatial coordinates.

In the hydrophone data in FIG. 4A, up-going primary energy 400 isobserved as a positive peak, and down-going ghost 402 is seen as anegative trough (the ghost observes a polarity reversal on reflection atthe sea surface). In the vertical particle velocity data in FIG. 4B,up-going primary energy 406 is also positive polarity; while ghost 408has still experienced a polarity reversal, it is now traveling downward.This “double reverse in polarity” means the ghost energy 408 is now alsoobserved as a peak. Results illustrated in FIGS. 4C and 4D show howup-going wave-field 410 and down-going wave-field 412 have beenaccurately separated by using one or more L′ transforms.

The above example shows wave-field reconstruction using hydrophone andvertical particle velocity data (i.e., the 2D case). However, it ispossible to extend matrix L of equation (5) to the 3D case, which mayalso use, for example, the horizontal particle velocity data, e.g., theparticle velocity component perpendicular to the streamers to extend theprevious example to wave-field separation and interpolationperpendicular to the streamers. The horizontal particle velocityinformation perpendicular to the streamer (also called cross-lineparticle velocity component) helps constrain the p_(y) direction ofmodel p to go beyond the point of natural aliasing that would beobserved with hydrophone-only data. In this case, model p becomes atau-p_(x)-p_(y) model which reverse transforms with transform L tosimultaneously satisfy hydrophone (h), vertical particle velocity(v_(z)) and horizontal particle velocity perpendicular to the streamers(v_(y)). It is also possible to extend the formulation for v_(x)measurements (particle velocity along the streamer direction, i.e.,inline particle velocity component) if available. In this case, eachelement in the tau-p_(x)-p_(y) model vector relates to slownesses in thex (parallel to the streamer) and y (perpendicular to the streamer)directions.

For example, for this case, the primary (up-going) and ghost(down-going) time delays for 3D wave propagation are defined as follows:

τ_(u)(n,m)=( h (n)+Δ h (n,m))· s (m)−Δτ(n,m)  (7)

τ_(d)(n,m)=( h (n)−Δ h (n,m))· s (m)+Δτ(n,m)  (8)

where h(n), Δh(n,m) and s(m) are now vector quantities relating tooffset-x/-y, delta-offset-x/-y, and slowness-x/-y, respectively. The dotproduct may be evaluated as:

( h (n)+Δ h (n,m))· s (m)=(h _(x)(n)+Δh _(x)(n))s _(x)(m)+(h _(y)(n)+Δh_(y)(n))s _(y)(m)  (9)

( h (n)+Δ h (n,m))· s (m)=(h _(x)(n)−Δh _(x)(n))s _(x)(m)+(h _(y)(n)−Δh_(y)(n))s _(y)(m)  (10)

The terms in equations (7), (8), (9) and (10) may be defined based onray geometry relating to FIG. 5A as follows:

$\begin{matrix}{{\sin \; {\underset{\_}{\alpha}(m)}} = {v_{w}{\underset{\_}{s}(m)}}} & (11) \\{{\Delta {\underset{\_}{h}\left( {n,m} \right)}} = {{z(n)}\tan \; {\underset{\_}{\alpha}(m)}}} & (12) \\{{\Delta {\tau \left( {n,m} \right)}} = \frac{\sqrt{{z^{2}(n)} + {{\Delta {\underset{\_}{h}\left( {n,m} \right)}}}^{2}}}{v_{w}}} & (13)\end{matrix}$

where v_(w) is the water velocity, α(m) is the vector incidence angle,z(n) is the depth of a given receiver, and Δτ(n,m) is the delta of thetime delay. The Δτ(n,m) quantity may be considered as the time betweenthe up-going ray passing through the receiver and reaching the seasurface. This time is the same as the time between the down-going rayleaving the sea surface and reaching the receiver as a ghost. This termis the 3D equivalent of the 2D terms in equations (3) and (4) which isdescribed in more detail in the '776 application.

Each trace will have its own delay based on the trace's individualoffset-x, offset-y and depth (i.e., its coordinates in the spacedomain). Note that because each trace has its own offset-x, offset-y anddepth, the L matrix also depends on the offset-x, offset-y and depth ofthe traces. Thus, as will be discussed later, an interpolation step maybe built in to the L′ matrix by selecting new offset-x and offset-y forthe output traces. Note that the interpolation step may be performedin-between the streamers, i.e., at any z relative to the z position ofthe streamers. In other words, the output traces noted above may beoutput in the same plane as the streamers or above or below this plane.In the interpolation step, it is also possible to output some receiversabove and some receivers below the streamers. A re-datuming step mayalso be built in to the L′ matrix in order to output at new receiverdepths. In this way, it would be possible to achieve, with the L′matrix, not only wavefield separation and/or interpolation, but alsoredatuming. Both redatuming and interpolation may be described by the“wave-field reconstruction” term. Thus, in the following, the termwave-field reconstruction includes wavefield separation (e.g.,deghosting), interpolation, or redatuming or combination thereof. Thesecapabilities of the L′ matrix are discussed later. The new transfermatrix L can be built from elements I_(u−d) and I_(u+d) defined asfollows:

I _(u−d) =e ^(−2πifτ) ^(u) −e ^(−2πifτ) ^(d)   (14)

I _(u+d) =e ^(−2πifτ) ^(u) −e ^(−2πifτ) ^(d) ,  (15)

where I_(u−d) is used for the hydrophone and horizontal particlevelocity terms, and I_(u+d) is used for vertical particle velocity data.By including obliquity terms we may define the 3D vector particle motionas follows:

$\begin{matrix}{{\underset{\_}{v}\left( {n,m} \right)} = \begin{pmatrix}{\sin \; \theta \; (m)\; \cos \; {\beta (m)}\; {l_{u - d}\left( {n,m} \right)}} \\{\sin \; \theta \; (m)\; \sin \; \beta \; (m)\; {l_{u - d}\left( {n,m} \right)}} \\{\cos \; \theta \; (m)\; {l_{u + d}\left( {n,m} \right)}}\end{pmatrix}} & (16) \\{{\sin \; \theta \; (m)} = {{\pm v_{w}}{{\underset{\_}{s}(m)}}}} & (17) \\{{{\tan {\beta (m)}} = \frac{s_{y}(m)}{s_{x}(m)}},} & (18)\end{matrix}$

where θ is the angle between the ray 510 and the vertical 512 and betais the angle between the surface projection 511 of the ray 510 (in theup-going surface offset-x/-y 506 and the x-axis 514 as shown in FIG. 5A.The acquisition system 500 illustrated in FIG. 5A includes a streamer502 and a single receiver 504 for simplicity. Receiver 504 is located ata given depth relative to surface 506, and, as discussed later, it maybe any type of receiver, e.g., single or multi-component.

Based on this 3D particle velocity, the matrix L can be adapted for 3Dray propagation as follows:

$\begin{matrix}{\begin{pmatrix}h_{1} \\h_{2} \\h_{N} \\{vz}_{1} \\{vz}_{2} \\{vz}_{N} \\{vy}_{1} \\{vy}_{2} \\{vy}_{N}\end{pmatrix} = {\begin{pmatrix}l_{u - d} \\{\cos \; {\theta (m)}\; {l_{u + d}\left( {n,m} \right)}} \\{\sin \; \theta \; (m)\; \sin \; \beta \; (m)\; {l_{u - d}\left( {n,m} \right)}}\end{pmatrix}\; {\begin{pmatrix}p_{1} \\p_{2} \\p_{3} \\p_{M}\end{pmatrix}.}}} & (19)\end{matrix}$

The top third of the L matrix and the top third of column vector data din equation (19) relate to hydrophone measurements, the middle third ofL and d relates to vertical particle velocity receiver data (V_(zi)),and the bottom third relates to horizontal particle velocity data(V_(yi)) perpendicular to the streamers. While this example uses thesame number of hydrophone, v_(z), and v_(y) components, other examplesmay have a different number of measurements for each component. Thus,the expressions “top third,” “middle third,” and “bottom third” usedherein should be understood in a liberal way, i.e., to include more orless than one third of the elements of the matrix or column vector data.Multi-component measurements may just be available for a limited rangeof offsets (e.g., near offsets) and hydrophone only measurementsthereafter. Some streamers may contain multi-component receivers, otherstreamers may be contain hydrophone only receivers. As before, model pis found to simultaneously satisfy hydrophone and particle velocitydata, i.e., as illustrated in FIG. 5B, when transform L is applied tomodel p, original seismic data d is obtained. In the case thathydrophone and particle velocity data have different bandwidths, afrequency dependent filter may be built into the equation as follows:

$\begin{matrix}{\begin{pmatrix}h_{1} \\h_{2} \\h_{N} \\{vz}_{1} \\{vz}_{2} \\{vz}_{N} \\{vy}_{1} \\{vy}_{2} \\{vy}_{N}\end{pmatrix} = {\begin{pmatrix}{l_{u - d}\left( {n,m} \right)} \\{F_{v}\cos \; {\theta (m)}\; {l_{u + d}\left( {n,m} \right)}} \\{F_{v}\sin \; \theta \; (m)\; \sin \; \beta \; (m)\; {l_{u - d}\left( {n,m} \right)}}\end{pmatrix}\; {\begin{pmatrix}p_{1} \\p_{2} \\p_{3} \\p_{M}\end{pmatrix}.}}} & (20)\end{matrix}$

This allows the derivation of a single model, p, which satisfieshydrophone data at all frequencies but is required to only satisfyparticle velocity data in a restricted frequency range, e.g., for higherfrequencies, e.g., above 30 Hz. While equation (20) uses a constantF_(v), in other examples this quantity may be permitted to vary for eachfrequency and also from trace to trace depending on noise characteristicvariations. In one application, different quantities may be used for thev_(z) and v_(y) components.

However, as also illustrated in FIG. 5B, after model p is found, anothertransform L′ may be applied to model p to obtain primary only, or ghostonly or another combination of data d′ as already discussed above. Newdata d′ may be at the same datum as the original data, i.e., z=z′, or atthe same exact positions, i.e., x=x′, y=y′ and z=z′, or it may beinterpolated, i.e., x or y or both may be different from x′ and y′, orit may be reconstructed at completely another position, i.e., x, y and zare different from x′, y′ and z′, respectively. New transform L′ maycontain up-going and down-going terms or also include other operations,i.e., deghosting, etc. In this document, the terms up-going and primaryare used interchangeably. The same is true for terms down-going andghost.

A synthetic dataset has been generated with pressure and particle motion(velocity) components as illustrated in FIGS. 6A-C. The datasetconsisted of 8 streamers with 75 m separation each and 12.5 m channelspacing. Other numbers may be used. FIGS. 6A-C show input data, and FIG.6D shows data output with up-going energy only. Lines 600 in FIGS. 4A-Dshow the streamer's depth variation (along axis 602) withsource-receiver offset (along axis 604). The presence of energy on thev_(y) component (FIG. 6C) indicates a cross-line component to theparticle motion. By including this or other components in the equation,better 3D deghosting may be achieved.

The v_(y) component is also useful for interpolation. In particular,properties of the v_(y) component are beneficial for reducing aliasing.FIGS. 7A-C show input synthetic data, and FIG. 7D shows results forcommon channel panels (streamers 1 to 8 for channel 1, followed bystreamers 1 to 8 for channel 21, etc. as illustrated in FIG. 7E) whenequation (19) is used.

It may be seen in FIG. 7F that the polarity of the v_(y) componentchanges for positive and negative y-offsets (X axis in the figure), asexpected. The input data in FIGS. 7A-C is displayed at 75 m spacing, butthe output up-going dataset in FIG. 7D has been output with 37.5 mstreamer spacing. This can be seen from the finer trace spacing on theoutput. Even though the input data is observed to be aliased, the outputis nicely reconstructed.

Thus, according to an embodiment, there is a method for processing inputseismic data d that includes a step of receiving input seismic data drecorded in a first domain by seismic receivers that travel in water,with input seismic data d including primary and surface ghostwave-fields; a step of generating a model p in a second domain todescribe input seismic data d; and a step of processing model p toobtain a seismic dataset indicative of ghost wave-fields andsubstantially free of primary wave-fields.

Input seismic data d includes only pressure measurements, or onlyparticle motion measurements, or both pressure and particle motionmeasurements. The first domain may be a time-space domain, while thesecond domain may be one of a Radon domain (hyperbolic, parabolic, etc),frequency-wave number domain, tau-p domain, rank reduction, singularvalue decomposition (SVD), and curvelet domain. In one application, thestep of generating a model p includes solving an inverse problem basedon linear operator L and the input seismic data d, and applying an L′transform to the model p to obtain a seismic dataset indicative of ghostwave-fields, with the L′ transform combining primary attenuation andinterpolation. The L′ transform may be applied after a denoising step isapplied to model p. In one application, an amount of noise is reduced bycontrolling sparseness weights when the model domain is derived. Thesparseness weights may also be derived to mitigate aliasing, which maybe especially useful if only hydrophone or only particle velocity datais input. The sparseness weights may be derived initially at lowfrequencies (e.g., at values less than 10 Hz) and used to constrain themodel at high frequencies. The sparseness weights may be updated duringseveral iterations, e.g., 0-10 Hz model is used to constrain a 0-20 Hzmodel which is used to constrain a 0-40 Hz model, etc. The sparsenessweights may be derived from the envelope of the tau-p model at eachiteration. Processing in the model domain may also include muting,scaling, resampling, removing cross-talk or interference noise,re-datuming and vector rotation, as will be discussed later.

The seismic dataset indicative of ghost wave-fields may be generated atpositions in-between the receivers, i.e., having any output z relativeto the zs of the receivers and/or streamers. The positions may be at adifferent datum than the receivers, or the positions are designed tomatch positions of receivers from another seismic survey, or thepositions are equidistant from input streamers on which the receiversare distributed, or the positions are on a regular grid.

In one application, the seismic dataset indicative of ghost wave-fieldsmay be subtracted from input seismic data d to obtain data d′ to be usedto generate a final image of a surveyed subsurface, or the seismicdataset indicative of ghost wave-fields is directly used to generate afinal image of a surveyed subsurface.

According to another embodiment, there is a method for processing inputseismic data d. The method includes a step of receiving input seismicdata d recorded, in a first domain, by seismic receivers that travel inwater, with the input seismic data d including pressure and particlemotion measurements; a step of generating a model p in a second domainto describe input seismic data d; and a step of processing model p tooutput a seismic dataset with attenuated noise. The step of processingmay include removing cross-talk noise in model p by muting, whereincross-talk noise is generated when two or more seismic sources generateseismic waves at the same time. The method may also include furtherremoving cross-talk noise based on the non-coherent nature of model p inthe second domain. An amount of noise may be reduced by controllingsparseness weights when an L transform is applied. In one application,the second domain is a common channel domain, and the first domain is atime-space domain. In another application, the second domain is one of aradon domain, frequency-wave number domain, rank reduction, SVD, tau-pdomain, and curvelet domain. For example, the data in the second domainmay be sorted into the common-p/shot domain in which the timing of thenoise may vary. Erratic/impulsive denoise methods may be used tomitigate the noise based on the inconsistent timing of the noise.

The step of generating a model p may include solving a linear problembased on input data and an operator L, followed by an L′ transform toobtain seismic dataset with attenuated noise, wherein the L′ transformcombines deghosting and interpolation. The L′ transform may also includeresampling, and/or redatuming and/or vector rotation. In oneapplication, the seismic dataset with attenuated noise is used togenerate a final image of a surveyed subsurface.

The above-discussed embodiments did not take into account theorientation of the receivers, which may change as the streamers aretowed in water. In other words, the receivers were assumed to beoriented parallel to the shooting direction, i.e., extending along theinline direction. However, often this is not the case because streamer800, when towed by vessel 802, may make a feather angle 804 withtraveling direction X (i.e., inline direction), as illustrated in FIG. 8and, thus, a receiver may be oriented in various ways relative to thestreamers.

To account for this, it may be necessary to modify the obliquity termsrelating to the v_(y) component on a trace-by-trace basis, based on theorientation of each receiver. In the following formulation, angle δ(n)represents the orientation of the streamer to the transform orientation(px-py) at the location of a given receiver 810 (which, for simplicity'ssake here, is defined as the nominal shooting direction). This isequivalent to the angle between the orientation of the v_(y) receiverand the model p_(y) direction. Thus, the modified form of the L matrixthat takes receiver orientation into account is given by:

$\begin{matrix}{\begin{pmatrix}h_{1} \\h_{2} \\h_{N} \\{vz}_{1} \\{vz}_{2} \\{vz}_{N} \\{vy}_{1} \\{vy}_{2} \\{vy}_{N}\end{pmatrix} = {\begin{pmatrix}{l_{u - d}\left( {n,m} \right)} \\{\cos \; {\theta (m)}\; {l_{u + d}\left( {n,m} \right)}} \\{\sin \; \theta \; (m)\; \sin \; \left( {{\beta \; (m)} + {\delta (n)}} \right)\; {l_{u - d}\left( {n,m} \right)}}\end{pmatrix}\; {\begin{pmatrix}p_{1} \\p_{2} \\p_{3} \\p_{M}\end{pmatrix}.}}} & (21)\end{matrix}$

While this embodiment is discussed in the context of particle velocitymeasurements in directions vertical and perpendicular (i.e., along across-line direction) to the streamer, the equations can be modified forparticle velocity receivers oriented in any direction. For example, dueto a streamer twisting while towed in water, accelerometers along thestreamer may be oriented in any direction. In addition, with avariable-depth streamer, it may be important to accurately model theorientation of the “vertical component” which may not be strictlyvertical, but rather perpendicular to the streamer.

Although the data may be preprocessed to reorient the recordings to avertical and streamer-perpendicular orientation, it is possible to leavethe data at the original orientation and design L matrix elements basedon the orientation of each individual receiver. While the discussionhere relates to a hydrophone plus two other components, it is possibleto modify the equations to any number of particle velocity measurementsoriented in any direction. The orientation of each receiver can bedifferent.

According to an embodiment, a method for processing input seismic data dincludes a step of receiving the input seismic data d recorded, in afirst domain, by seismic receivers that travel in water, with the inputseismic data d including pressure and particle motion measurements; astep of generating a model p in a second domain to describe the inputseismic data d, with model p taking into account an obliquity ofincoming wave-fields; and a step of processing model p to output aparticle motion dataset corrected for obliquity. The method may includewave-field reconstruction of incoming wave-fields based on pressure andparticle motion measurements and model p.

In one application, incoming wave-fields are reconstructed at newreceiver positions. The particle motion dataset corrected for obliquityincludes surface ghost wave-fields, or is substantially free of surfaceghost wave-fields. In one embodiment, the first domain istime-offset/x-offset/y-depth and the second domain istau-slowness/x-slowness/y, where offset/x is a distance between a sourcegenerating input seismic data d and a corresponding receiver along aninline direction, and offset/y is a distance between the source andcorresponding receiver along a cross-line direction, which issubstantially perpendicular on the inline direction.

In one application, the output particle motion dataset corrected forobliquity is not interpolated. In another application, a component ofthe particle motion dataset corrected for obliquity is summed withpressure data in the first domain to obtain wave-field separation.Similar to other methods discussed herein, an L′ transform may beapplied to obtain the particle motion dataset corrected for obliquity,wherein the L′ transform combines the obliquity and wave-fieldreconstruction. The L′ transform may include other operations as alreadydiscussed.

According to another embodiment, there is a method for processing inputseismic data d that includes receiving the input seismic data drecorded, in a first domain, by seismic receivers that travel in water,with input seismic data d including pressure and particle motionmeasurements; generating a model p in a second domain to describe inputseismic data d; and processing model p to output a seismic dataset withattenuated noise. The processing step may include removing cross-talknoise in model p by muting, wherein cross-talk noise is generated whentwo or more seismic sources generate seismic waves at the same time.

According to still another embodiment, it is possible to derive a singlefixed datum model which, when reverse-transformed, simultaneously modelsall available particle velocity and hydrophone measurements (in thiscase hydrophone, vertical v_(z) particle velocity receiver, and v_(y)horizontal particle velocity receiver perpendicular to the streamers;inline v_(x) may be added) at the recording depths and offset-x/offset-ypositions.

Once the tau-px-py model has been found (no matter which formulation orwhich components have been used), it can be used to output up-goingdata, down-going data, or a combination of the two at any spatialposition, where the term “spatial position” means x (inline position), y(cross-line position), and z (receiver depth) coordinates. Note that theterm “output” includes one or more of wavefield regularization,interpolation, deghosting, denoising, redatuming, resampling, etc. Eachindividual output trace can have its own x, y, z location (e.g., generalfloating datum). This location may be at the position of the originalreceivers, in between the original receivers, for example, interpolationat a new streamer location, at any receiver depth, or a combination ofin between the streamers at different depths. This method can be used tooutput up-going energy at the surface or other horizontal datum, whichmay be subsequently processed with conventional processing algorithms.The method can also be used to output down-going energy at the surfaceor any other datum, and again, this may be processed with conventionalalgorithms.

The interpolation aspect of this method can be used to de-alias thecross-line sampling of a dataset, or to map on to the exact receiverx-y-z coordinates of another dataset, towed streamer, OBS, or landdataset. The data may be output for up-going, down-going or acombination of the two. The output positions of all traces can be at anyx-y-z coordinate within the original streamer spread or outside thespread (extrapolation). Up-going and down-going datasets may be outputat different datums if required.

In another embodiment, the L matrix discussed above may be used fortime-lapse studies where one or more vintage datasets consist ofmeasurements at different spatial coordinates and/or receiver depthsthan new acquisition measurements. Once the model p has been found, itmay be used to output data at the exact x-y coordinates and depths ofany prior vintage (baseline) dataset or other positions. This allowsaccurate comparison of vintage datasets and reconstructed monitordatasets. Up-going, down-going or a combination of both may be used forthis purpose. For example, a base hydrophone-only dataset will containprimary and ghost data, and interpolation or deghosting of this basedataset may not be possible. In this case, it can be of interest tooutput the monitor data (the later-in-time survey data) at the x-y-zrecording coordinates of the baseline, including primary and ghost. Withmultiple datasets, it may be of interest to interpolate all vintages onto a common sampling that includes positions not occupied by anydataset. The positions could be designed so that the interpolationdistance on average is minimum, i.e., the positions are selected asclose as possible to the input data positions because the interpolationquality at positions further away is expected to degrade.

According to another embodiment, different x/y offsets and depths may beused for up-going and down-going datasets, for example, to improveillumination or to match wave-field propagation to a vintage dataset ordatasets.

The scheme may be used to output particle velocity data onto a secondset of traces to help interpolation, e.g., if a first base datasetincludes only hydrophone data and a monitor dataset includesmulti-component data, it is possible to interpolate the monitor datasetto the positions of the base and then use the base hydrophone combinedwith interpolated monitor particle velocity for interpolation of thebase hydrophone data. One example is the use of a monitorhydrophone/particle motion dataset for outputting particle velocitymeasurements on a vintage dataset. Interpolated particle velocitymeasurements combined with original pressure measurements from thevintage data can be used for interpolation of the vintage dataset.Alternatively, particle motion data may be extrapolated within a shotgather from near offsets (where accelerometer measurements areavailable) to far offsets (where accelerometers were not installed).

The methods discussed in the previous embodiments can be generalized toa method that derives a fixed datum model to satisfy any number ofparticle velocity data with any 3D orientation, along with a differentnumber of hydrophone measurements. Measurements of hydrophone andparticle motion need not be co-located in space or depth. Individualparticle motion receivers need not be consistently oriented.

In this regard, consider N hydrophone measurements and O particlevelocity measurements. Hydrophone and particle velocity measurementseach have their unique coordinates (x, y, and z (depth)). First, referto equations (16) to (18) describing the particle velocity vector basedon a known model:

$\begin{matrix}{{\underset{\_}{v}\left( {n,m} \right)} = \begin{pmatrix}{\sin \; \theta \; (m)\; \cos \; {\beta (m)}\; {l_{u - d}\left( {n,m} \right)}} \\{\sin \; \theta \; (m)\; \sin \; \beta \; (m)\; {l_{u - d}\left( {n,m} \right)}} \\{\cos \; \theta \; (m)\; {l_{u + d}\left( {n,m} \right)}}\end{pmatrix}} & (22)\end{matrix}$

using the angle definitions introduced in FIG. 5A.

The three particle velocity components can then be re-oriented to theorientation of each individual receiver. Receiver orientation may varyfrom shot to shot, and even during the recording time of a trace. Oneway to re-orient the receiver is to take the resolved component of thewave-field vector through use of the dot product. This possibility isnow discussed in more detail. However, other ways may be imagined bythose skilled in the art.

If v is a vector having components (v_(x), v_(y), v_(z)) representingthe three particle motion components for a given receiver, and unitvector a(|a|=1) represents the receiver orientation relative to thetransform orientation (p_(x), p_(y) as in FIG. 8), the following maydescribe the resolved component of the wave-field along the receiverorientation:

a·v=|a∥v|cos σ.  (23)

The following notation may be introduced:

r=|v |cos σ  (24)

in which case r may be rewritten as:

$\begin{matrix}{{\cos \sigma} = {\frac{r}{\underset{\_}{v}} = \frac{\underset{\_}{a} \cdot \underset{\_}{v}}{{\underset{\_}{a}}{\underset{\_}{v}}}}} & (25) \\{r = \frac{\underset{\_}{a} \cdot \underset{\_}{v}}{\underset{\_}{a}}} & (26) \\{{r = {\frac{{a_{x}v_{x}} + {a_{y}v_{y}} + {a_{z}v_{z}}}{\underset{\_}{a}} = {{a_{x}v_{x}} + {a_{y}v_{y}} + {a_{z}v_{z}}}}},} & (27)\end{matrix}$

where σ is the angle between wave-field vector v and receiverorientation a. Wave-field vectors v vary with p_(x) and p_(y), and thereceiver orientation a varies with the shot and time for each receiver.As noted above, other re-orientation schemes can be used.

Based on equation (27), the final matrix formulation may be given by:

$\begin{matrix}{\begin{pmatrix}h_{1} \\h_{2} \\h_{N} \\v_{1} \\v_{2} \\v_{3} \\v_{4} \\v_{5} \\v_{0}\end{pmatrix} = {\begin{pmatrix}l_{a} \\r_{1} \\r_{2} \\{\; r_{0}}\end{pmatrix}\; \begin{pmatrix}p_{1} \\p_{2} \\p_{3} \\p_{M}\end{pmatrix}}} & (28)\end{matrix}$

where the top part of the L matrix and the top part of data vector d(receivers 1 to N) relate to hydrophone measurements, and the followingrows for L matrix and data vector d relate to particle velocity data(receivers 1 to O). Model vector p contains M values. In one embodiment,M values include slownesses in the x and y directions, although othermodel domains may be used as discussed later.

The transform L from the coordinate system of model p to the receivers'coordinate system (e.g., time-space domain) may vary for each receiverand with time. This means the orientation of the receivers can vary fromshot to shot, but also during recording of an individual trace. The L′transform may be designed or coded based on equation (27), for eachreceiver, to rotate the particle motion receivers (i.e., theirrecording) to be aligned with the y-axis, z axis, or any preferred axis.This step of vector rotation may be applied to any of the embodimentsdiscussed herein. In this way, the input particle motion measurementsmay be received with arbitrary orientation, being allowed to vary foreach receiver and are not substantially orientated in two or fewerdirections. Output particle motion measurements may be formed such thatthey may also take any arbitrary orientation. The orientation of theoutput particle motion receivers may be the same as the input particlemotion receivers or may be different to the input particle motionreceivers. The term receiver orientation should relate to theorientation of the streamer data received by the processing algorithm.This orientation may be different to the orientation of the originalreceiver if the data has previously being re-orientated. Severalindividual receivers may be summed to form a receiver group. Thereceiver data input to the algorithm may relate to individual receiversor receiver groups. In other words, there is a method for processinginput seismic data d that includes a step of receiving the input seismicdata d recorded in a first domain by seismic receivers that travel inwater, the input seismic data d including pressure and particle motionmeasurements, a step of generating a model p in a second domain todescribe the input seismic data d, and a step of processing the model pto obtain an output seismic dataset, the output seismic datasetincluding wave-field reconstruction of the pressure wave-fields and/orparticle motion wave-fields based on the model p. The particle motionmeasurement's orientations may vary from receiver to receiver. In oneapplication, the particle motion measurement's orientations vary betweengroups of receivers. In another application, part of all theorientations are not along predetermined directions, e.g., y and zdirections.

The above-discussed transforms may be adapted to work under differentconditions. For example, through reciprocity, any of the above methodscan be modified to work in the common receiver domain where there is asufficient sampling of shots. This can be of use for land or OBSdatasets where there is a need to redatum, deghost, reghost, etc.

The 3D algorithms introduced above may be used for many things, someexamples are given below:

-   -   1) Output up-going (primary) data at the streamer datum. The        primary may be subsequently subtracted from original pressure        data to leave down-going energy only.    -   2) Output down-going (ghost) data at the streamer datum. The        ghost may be subsequently subtracted from original pressure data        to leave up-going energy only.    -   3) Output up-going or down-going data at surface datum.    -   4) Output up-going or down-going data at a new horizontal datum.        This allows deghosted data to be provided at the start of the        processing sequence, so that the data can be processed with        conventional algorithms.    -   5) Output up-going or down-going data at a new variable depth        datum.    -   6) Output up-going and down-going data at a new datum. For        example, this can be useful for time-lapse studies.    -   7) It is also possible to apply deghosting and redatuming in two        independent steps. First deghost the data (leaving it at the        original depths), then redatum it as a separate operation.    -   8) Instead of outputting up-going and down-going wave-fields, it        is possible to output particle motion data at any x-y-z location        with any orientation. This allows the output of particle motion        data on or between the streamers that is substantially free of        down-going (ghost) or up-going (primary) energy. The output        particle motion data at or between the streamers may or may not        be corrected for obliquity. This data may be processed in        parallel with pressure data and combined post-migration with a        joint deconvolution-type deghosting approach. Alternatively, it        may be summed with pressure data to achieve wavefield        separation.    -   9) In addition, particularly with the use of sparse inversion,        this procedure can be used for joint interpolation, deghost and        denoise.    -   10) The tau-p_(x)-p_(y) model can be used to separate cross-talk        noise (with simultaneous shooting) or interference noise based        on model parameter ranges.        -   a. In simultaneous shooting, the wave-field from two sources            is recorded;        -   b. Where the sources are at different spatial positions,            there may be a difference in the angle at which the            wave-field approaches the receivers;        -   c. When this is the case, the wave-fields may be denoised by            muting the noise in the tau-px-py domain. If the separation            is not perfect, the noise may be further separated based on            the non-coherent nature of the data in a given domain, for            example, the common p/shot domain;        -   d. Subsequently, a reverse tau-px-py transform may be used            to separate energy from the different sources in combination            with any other items listed here, or to come back to the            original coordinates.    -   11) It is possible to make use of the model domain p for        demultiple purposes, after which data (any combination of the        above) may be output on the input sampling or elsewhere, with or        without free surface ghost. One example is the application of        multiple attenuation using tau-px-py deconvolution in the model        space. In the tau-px-py domain (model domain p) the multiples        may be identified based on velocity discrimination and/or        periodicity. Multiples are periodic in model domain p. Thus,        recorded data d may be transformed to the model, autocorrelation        functions may be calculated for each trace, a lag is determined        based on autocorrelations for, e.g., predictive deconvolution,        the deconvolution is applied to select the multiples, and the        multiples are then transformed back to the time-space model        where they are removed from original data d. Normally, sampling        restrictions mean that tau-p deconvolution is applied in 2D in        either shot or receiver domain (or both). The results may be        sub-optimal due to the data exhibiting 3D wave field propagation        in reality. The use of particle motion measurements to make a 3D        (tau-p_(x)-p_(y)) model p can thus lead to improved demultiple.    -   12) Any combination of the above-noted embodiments or parts of        them may also be envisioned by one skilled in the art.        The above options can be used to output data at the same offsets        as input data, or at new offsets. These are only a set of        options, and those skilled in the art could easily imagine other        options based on the above-discussed concepts.

It can be also possible to extend the above methods to work on shot andreceiver deghosting and redatuming simultaneously. This could beconsidered as an algorithm with 7D input, i.e., source at positionsS(x,y,z) and receivers at positions R(x,y,z). Receivers can include amixture of hydrophones and particle motions sensors (as will bedescribed later), with each particle motion sensor having its ownorientation a₀(x,y,z) as before. This arrangement could lead to a 5Dsurface consistent model tau-p_(sx)-p_(sy)-p_(rx)-p_(ry) where p_(sx)and p_(sy) are source slownesses in the x and y directions, and p_(rx)and p_(ry) are receiver slownesses in the x and y directions. Wheresufficient sampling of the source is not available, it would be possibleto reduce the number of dimensions to S(x,z) and R(x,y,z), where thesources are positioned along a 2D line. Sparseness constraints can beused to help with interpolation beyond aliasing. In addition, sparsenesscan be used to denoise. Further, this algorithm or any of the methodsdiscussed herein may be combined with source interpolation for obtainingwavefield reconstruction on the receiver side.

In addition, source directivity compensation may be added in to theabove formulation by including a resignature operation as a function ofsource takeoff slowness. The resignature operation may include sourceairgun array effects as well as the source ghost. Assumptions relatingto source and receiver takeoff slownesses may be made based on raytracing, which may be in 1D (e.g., normal moveout (NMO)) and use acomplex velocity model.

Input data for any of the above methods may be in any pre-stack domain,for example shot, receiver, midpoint, conversion point or cross-spread.The intention is that any of the above implementations would be made ona computer. While much of the previous embodiments discussed usemulti-component measurements, it should be noted that wheresignal-to-noise ratio and sampling allows, the scheme(s) may be usedwith fewer data, e.g., hydrophone data only or particle motion dataonly. In particular, this may require more demands on sparsenessconstraints, e.g., beginning by solving the equations for a lowfrequency bandwidth which is not aliased, and using the model to derivesparseness weights for the higher frequency model solution. Also, it maybe possible to use as input pressure and particle motion data and togenerate an output that includes only pressure wave-fields or onlyparticle motion wave-fields, as now discussed.

One embodiment may include using hydrophone only data to construct anestimate of particle motion data. The estimate of particle motion datamay or may not be corrected for obliquity. The estimate of particlemotion data may or may not include wavefield separation. The estimate ofparticle motion data may be subtracted from recorded particle motiondata to leave a residual. This residual particle motion data may then bedenoised following which the estimated particle motion data is added tothe resulting denoised data.

For example, in one embodiment there is a method for processing inputseismic data d that includes a step of receiving input seismic data drecorded in a first domain by seismic receivers that travel in water,with the input seismic data d including pressure and particle motionmeasurements representative of primary and surface ghost wave-fields; astep of generating a model p in a second domain to describe inputseismic data d, wherein there is a mathematical transform L which, whenapplied to model p, results in input seismic data d; and a step ofprocessing with a mathematical transform L′ the model p to obtain, inthe first domain, a seismic dataset indicative of pressure wave-fieldsand having a characteristic imparted by transform L′, whereinmathematical transform L′ is different from mathematical transform L.

The L′ transform may be obtained in various ways, for example, byselecting one or more parameters in the L transform to be zero. In oneapplication, the L′ transform has more terms than the L transform. Thecharacteristic is related to the pressure wave-fields being interpolatedamong receiver positions and being free of primary wave-fields, or tothe pressure wave-fields being deghosted and interpolated among receiverpositions, or to the pressure wave-fields being interpolated amongreceiver positions based on rotated particle motion measurements, or tothe pressure wave-fields being interpolated along streamers includingthe seismic receivers, or to the pressure wave-fields being calculatedat a new datum relative to a datum of the seismic receivers. Thepressure wave-fields may be spatially resampled at another new datum. Inone application, the characteristic is related to the pressurewave-fields being deghosted and calculated at a new datum relative to adatum of the seismic receivers.

The method may also include denoising the model p prior to applying theL′ transform. In one application, the characteristic is related to thepressure wave-fields being interpolated at positions designed to matchpositions of receivers from another seismic survey. The positions may beequidistant from input streamers on which receivers are distributed, orthey may be on a regular grid. The first domain may be a time-spacedomain, and the second domain may be one of a radon, tau-p,frequency-wave number, SVD, rank reduction, tau-p and curvelet domain.

According to another embodiment, it is possible to implement a methodfor processing input seismic data d that outputs only particle motion oronly hydrophone recordings. The method includes a step of receivinginput seismic data d recorded in a first domain by seismic receiversthat travel in water, with the input seismic data d including pressureand particle motion measurements; a step of generating a model p in asecond domain to describe input seismic data d, wherein there is amathematical transform L which, when applied to model p, results ininput seismic data d; and a step of processing with a mathematicaltransform L′ the model p to obtain, in the first domain, a seismicdataset indicative of particle motion wave-fields. Mathematicaltransform L′ is different from mathematical transform L. In oneapplication, the step of generating a model is based on an L transformwhich incorporates a differentiation in time or space. Thisdifferentiation step converts particle velocity data to particleacceleration, or particle displacement data to particle velocity data,or particle velocity data to pressure gradient data, etc. Thedifferentiation may be applied in the time domain or the frequencydomain and may be a differentiation in time or in space. In anotherapplication, the step of generating a model is based on an L transformwhich includes a frequency filter term to account for different signalto noise ratios for pressure and particle motion data.

In one application, the seismic dataset indicative of particle motionwave-fields is interpolated among receiver positions and is free ofsurface ghost wave-fields. The L transform may include vector rotationcorrections for particle motion measurements which allows each particlemotion receiver to take its own unique orientation. The seismic datasetindicative of particle motion wave-fields may be calculated at a newdatum relative to a datum of the seismic receivers, or the seismicdataset indicative of particle motion wave-fields is spatially resampledat another new datum. In one application, the seismic dataset indicativeof particle motion wave-fields is deghosted and calculated at a newdatum relative to a datum of the seismic receivers.

In one application, the method includes a step of denoising model pprior to applying the L′ transform. The seismic dataset indicative ofparticle motion wave-fields may be interpolated at positions designed tomatch positions of receivers from another seismic survey. The positionsmay be equidistant from input streamers on which receivers aredistributed, or may be on a regular grid. In one application, theseismic dataset indicative of particle motion wave-fields is used forgenerating a final image of a surveyed subsurface.

Another embodiment that takes advantage of the L matrices discussedabove relates to the use of multicomponent measurements for combineddeghosting and redatuming. Fewer component measurements may be usedwhere S/N ratio and sampling allow, e.g., hydrophone only or particlemotion only. The strategy described next is aimed at improving spatialimage resolution in the image domain (after migration).

For illustrating the combined deghosting and redatuming, FIG. 9 shows anacquisition system 900 that includes streamers 902 with 100 m separationtraveling into the page. For a given cross-line slowness propagationdirection, there is a lateral shift Δh with which rays 906 reach the seasurface 908 relative to the streamers, and this separation is given by:

$\begin{matrix}{{\sin \; \theta} = {{vp} = \frac{\Delta \; h}{\sqrt{{\Delta \; h^{2}} + z^{2}}}}} & (29) \\{{{\Delta \; h} = \frac{vpz}{\sqrt{1 - {v^{2}p^{2}}}}},} & (30)\end{matrix}$

where p is the slowness (s/m) of a given wavefront and v is the watervelocity.

Even though there is a lateral shift Δh, it should be noted that thespatial sampling of the up-going energy (receivers below the seasurface) is the same at the sea surface as it is at the streamers' level(in this case 100 m). Considering now the sampling of the ghostwave-field, the concept of “mirror-receivers” may be used to positionmirror-streamers 910 having receivers above the sea surface 908 abovethe actual receivers streamers as illustrated in FIG. 9, which showsthat the surface spatial shift of rays 912 for down-going (ghost) datahas opposite polarity to that of up-going (primary) rays 906.

Based on this observation, according to an embodiment, it is possible toincrease effective spatial sampling without interpolating data inbetween the streamers. For example, consider the case in which single ormulticomponent data is available. This data may be used to jointlydeghost and redatum to a plurality of different depths, thus increasingthe amount of data. If, for example, it is desired to increase six-foldthe spatial resolution for a given dip, it is possible to choose sixhorizontal datums 1000-1010 for the output data as shown in FIG. 10.While FIG. 10 illustrates this concept with receivers below (associatedwith the primaries) and above (associated with the ghosts) the seasurface, the same can be achieved with receivers only below the surfaceas shown in FIG. 11.

The previous illustrative example describes pictorially how spatialresolution may be improved by redatuming the data to a plurality ofrecording depths. In one embodiment, this redatuming is combined withdeghosting. Thus, ghost receiver data should be prepared to have thesame polarity as receivers below the water surface. However, inpractical situations, it may be attractive to reconstruct receiversabove the ocean bottom, especially in the case of shallow water depthsor where there is a strong water velocity variation.

Next, some practicalities of this concept are considered based on anexample. Consider a survey with maximum cross-line dip (slowness) of0.0005 s/m, and an increase from 100 m sampling to 12.5 m sampling inspatial resolution is desired. Rearranging equation (30) as noted belowin equations (31) to (33), it can be seen that the increase in spatialresolution may be achieved through a depth increment of 11 m as follows:

$\begin{matrix}{{\Delta z} = \frac{\Delta h\sqrt{1 - {v^{2}p^{2}}}}{vp}} & (31) \\{{\Delta \; z} = \frac{1{2.5}\sqrt{1 - {1500^{2} \times 0.0005^{2}}}}{1500 \times {0.0}005}} & (32) \\{z = {1{1.0}2\; m}} & (33)\end{matrix}$

i.e., the deghosted virtual streamers' output is calculated at 0 m, 11m, 22 m, 33 m, 44 m, 55 m, 66 m and 77 m. In other words, for fixedstreamer spacing, the virtual streamer depth sampling is a function ofwater velocity, required horizontal resolution (Δh) and maximum slowness(p). Curve 1200 of the graph illustrated in FIG. 12 describes thisrelationship.

Curve 1200 seems unfavorable for low slownesses because the depthincrement increases rapidly. However, it should be noted that effectivespatial sampling for low frequencies need not be as fine as for highfrequencies. By definition, energy with vertical propagation does notvary laterally. This concept may be defined by spatial aliasing and theFresnel zone.

The frequency at which data spatially aliases may be given by thefollowing equation:

$\begin{matrix}{{f\; \max} = {\frac{v}{2{\Delta x}\; \sin \; \theta} = {\frac{v}{2\Delta \; {xvp}} = \frac{1}{2\Delta \; {xp}}}}} & (34)\end{matrix}$

where v=water velocity (m/s); Δx=spatial sampling, e.g., streamerspacing (m); and p=slowness (s/m). Rearranging equation (34) fordelta-x, the spatial sampling is found as a function of slowness:

$\begin{matrix}{{\Delta \; x} = {\frac{1}{2\; {pf}\; \max}.}} & (35)\end{matrix}$

For 100 Hz maximum frequency, the spatial sampling required to avoidaliasing and the streamer depth-increment may be calculated as shown inTable 1.

TABLE 1 Spatial Depth Slowness sampling increment (s/m) (m) (m) 0.00068.333333 4.315348 0.00055 9.090909 6.487211 0.0005 10 9.089282 0.0004511.11111 12.44497 0.0004 12.5 17.01727 0.00035 14.28571 23.5902 0.000316.66667 33.63464 0.00025 20 50.21793 0.0002 25 80.67466 0.0001533.33333 146.4034 1E−04 50 334.1174 5E−05 100 1347.646

Note that, in practice, effective spatial sampling for migration mayoften be finer than this by a factor of 2 for a pair of sources (flipand flop sources used routinely with dual-source single vesselacquisition). However, in practice it is necessary to go down to a depthof about 150 m.

While FIG. 10 shows consistent spatial sampling for a given slowness,for other slownesses with the same depths this will not be the case, asillustrated in FIG. 13. In this case can be seen a fine spatial sampling1302 followed by a gap 1304. As long as the gap is smaller than thealiasing condition for the given dip, the resolution will still besufficient. While it is possible to continue the fine receiver depthsampling deeper (almost to have a “curtain streamer”) to fill thesurface sampling, it can be more efficient to have a varying streamerdepth sampling because the sampling gets coarser with depth.

In this respect, FIGS. 14A-G illustrate (vertical axis corresponds todepth, and horizontal axis corresponds to inline direction) left-handside dots aligned vertically that relate to mirror streamer positionsfor streamer 1, and right-hand side dots that relate to streamerpositions for streamer 2. Streamer depths are as follows: 0 m, 5 m, 10m, 15 m, 20 m, 25 m, 35 m, 45 m, 55 m, 70 m, 90 m, 150 m. Surfaceprojections of different plane waves recorded by the receiversillustrated in FIGS. 14A-G show that the surface spatial sampling willbe sufficient for the spatial resolution required for all plane waves,based on aliasing figures quoted in Table 1.

Another alternative is to have two “curtain streamers” at either edge ofthe streamer spread. Although this solution may seem to relate to adegradation of spatial resolution, however, based on the schemesdiscussed herein, it may be appreciated that a similar spatialresolution may be obtained.

For a densely spatially sampled dataset on a horizontal datum, it iswell known that by transforming the data into the tau-px-py domain andapplying phase shifts for each slowness trace, the input datum may bemoved to another fixed level, i.e.,

-   -   1. Receive data in the (t,x,y) domain;    -   2. Transform it to the (tau,p_(x),p_(y)) domain;    -   3. Apply phase (time) shifts for each slowness trace (i.e., same        or different shifts trace by trace); and    -   4. Reverse transform the data to the (t,x,y) domain.

For a single curtain receiver, a similar process may be implemented,i.e.:

-   -   1. Receive data in the (t,y,z) domain;    -   2. Transform it into the (tau,p_(y),p_(z)) domain;    -   3. Apply phase (time) shifts for each slowness trace (i.e., same        or different shifts trace by trace); and    -   4. Reverse transform to the (t,y,z) domain.

In the first case, data at different receiver depths is generated usingthe (tau,p_(x),p_(y)) domain, while in the second case data is generatedat different y-positions (i.e., lateral sampling) using the(tau,p_(y),p_(z)) domain. Therefore, instead of spatially interpolatingdata and performing the migration, it is possible to extrapolate thedata to different receiver depths and then migrate it. In other words,it is possible to receive input seismic data while in tow, use the inputseismic data to generate data at a plurality of different receiverdepths, and then image the subsurface. The input data may includehydrophone data, or particle velocity data, or both. In one application,the data at different receiver depths is substantially free of freesurface ghosts (down-going energy) or is substantially free of primaryenergy (up-going energy). Receivers may be below or above the freesurface, they may be provided at receiver depths that are regularly ornot spaced in depth, and/or the receiver depths may be designed tooptimize spatial resolution at the sea surface or along subsurfacestructures of interest. Models in p_(x), p_(y), and p_(z) may be linkedthrough the inverse square of the water velocity through equation:v⁻²=p_(x) ²+p_(y) ²+p_(z) ².

The following comments relate to the design and use of the L matrixdiscussed above. Particle velocity data may be obtained from individualsensors, or summed (average or weighed sum) to form a receiver group.Particle velocity data may have been acquired directly or may becomputed from accelerometer sensors (for example, by integration). Othertypes of particle motion sensor may be available. While the aboveembodiments relate to modeling of particle velocity data, adifferentiation step may be included in the matrix formulations to workdirectly with accelerometer data. The differentiation could be appliedin the time or the frequency domain. Receivers generate a marinestreamer dataset that is achieved in a narrow, wide or multi-azimuth,coil shooting or any configuration towed with constant or variable depth(e.g., slant streamer, BroadSeis profile, over-under streamers), and theseismic data is generated with an air gun, marine vibrator, or othersource element. Source elements may be fired according to any knownscheme, e.g., continuously, simultaneously, flip-flop, etc. Receiversmay also be used in ocean bottom survey (nodes, cables, or other withair gun, marine vibrator or other source), land dataset (dynamite,vibrator or other source), or a combination of two or more datasettypes. The data may have been calibrated before applying the processesdiscussed herein, or calibration scalars may be included in the matrixformulations noted in the embodiments. Water velocity terms may beconstant or allowed to vary with depth. Variation with depth can be ofuse for OBS datasets where there is a water velocity gradient. Themethods may be used for one-sided or split-spread acquisition.

Equations described herein may be solved in the time domain or aspectral domain (e.g., frequency, Laplace, z-transform, etc.), waveletdomain (e.g., curvelet or other). Model p may be found through anyinversion method, e.g., conjugate gradients, LU decomposition, Choleskyfactorization, etc. Model p may be derived to represent all traces inthe input shot, or may work on a subset of data from the input shot, forexample, spatial windows of a given number of channels. Sparsenessweights may be used in the inversion to improve results, for example,where there is poor signal-to-noise ratio or to overcome aliasing; e.g.iteratively reweighted least squares beginning with low frequencies andworking up to higher frequencies. Other model domains may be used, forexample, frequency-wavenumber (FK), parabolic Radon, hyperbolic Radon,etc. In fact, any fixed datum model domain may be defined as long as itcan be reverse transformed, redatumed and reghosted for both hydrophoneand particle velocity sensor data. Alternatively, an iterative approachsimilar to the anti-leakage tau-p transform can be used which alsoexhibits sparseness properties. No matter how the model is formed, itneeds to simultaneously reproduce the hydrophone and particle velocitymeasurements through application of an operator, e.g., L.

Due to differing signal to noise ratio of hydrophone and particlevelocity data, it may be necessary to define the inversion so as tosatisfy the hydrophone data for a broader bandwidth than the particlevelocity data. This may be implemented by including a frequencydependent scaling term into the matrix or bandpass filtering the modeland data for different conjugate gradient passes either bymultiplication in the frequency domain or convolution by a bandpassfilter in the time domain. For example, application of L may include abandpass filter so that when applied the bandwidth of particle velocitycomponents is 25 Hz to 250 Hz, whereas the bandpass filter forhydrophone data is 2 Hz to 250 Hz. Conjugate gradient inversion beginsby computing L^(T)d from d, and continues by combining frequencyfiltering into L. The bandwidth of L^(T)d will automatically be adjustedand be consistent for the later applications of L and L^(T) in theconjugate gradient flow.

It can also be possible to process hydrophone and particle motion dataindependently. The separate results may be combined afterwards, forexample, by selecting different temporal frequency ranges based onsignal-to-noise ratio data. At low frequencies, particle velocity datamay be too noisy to be of value. Therefore, it may be necessary to use apressure-only solution for low frequencies, and make use of a combinedhydrophone-particle velocity solution for higher frequencies.

A generalized weighting scheme can be implemented to vary weightingbetween any component (hydrophone or particle motion) depending on thereceiver, time and frequency and/or wavenumber. This weighting refers tohow well the model represents the data. Typically, accelerometer data isintegrated to calculate particle velocity measurements. Instead ofintegrating accelerometer data before wave-field separation, it is alsopossible to build a differentiation operator into the inversion scheme.In the time domain, the application of operator L would then includeredatum, reghost and differentiation. As an alternative to the describedmodeling approach, other forms of wave-field extrapolation may be used,which may include Kirchhoff, beam, wave-equation (one-way or two-way,RTM, etc).

The embodiments discussed above with regard to determining the L matrixhave focused on illustrating how the mathematics works and how thevarious steps, e.g., deghosting, datuming, interpolation, denoising,etc. are achieved by modifying the L matrix. The next embodimentsdescribe possible implementations of these steps according to methodsthat are now discussed.

According to one embodiment, it is possible to design the L matrix sothat joint interpolation, deghosting and/or denoising is achieved usingmulticomponent streamer measurements. The method may include, asillustrated in FIG. 15, a step 1500 of receiving particle motion dataand pressure data from seismic receivers towed by a vessel. Thereceivers may be part of a plurality of streamers, ocean bottom cable orautonomous underwater vehicles. The pressure and particle motion dataincludes surface ghosts. The method also includes a step 1502 ofprocessing particle motion data and pressure data to generate a data setindicative of a denoised pressure wavefield, at positions between thestreamers and substantially free of surface ghosts. To achieve thisprocessing step, the model p is determined, then, in the model domain,the noise is muted and then an L′ matrix is applied to reverse transformthe data, deghost and interpolate. Prior or after the step of muting thenoise, the model p may be used for wavefield separation. The L′ matrixis designed to produce new offset-x and offset-y positions to take careof the interpolation, and also to remove surface ghosts. Thus, the datad′ obtained in the time domain as a result of the L′ matrix, which mayinclude only surface ghosts, is subtracted from original data d toobtain only primaries.

In one application, if the accelerometer data is noisy, which tends tobe inherent particularly at low frequencies, instead of first denoisingthe data in the p model, followed by interpolation and deghosting withmatrix L′, the whole process may be integrated into the L′ matrix thanksto the use of sparseness weights.

According to another embodiment, joint interpolation and deghostingusing multicomponent streamer measurements may be implemented without adenoising step as discussed in the embodiment of FIG. 15. Thus, for thismethod, as illustrated in FIG. 16, there is a step 1600 of receivingparticle motion data and pressure data from seismic receivers that aretowed, and the pressure and particle motion data includes surfaceghosts. A next step 1602 processes particle motion data and pressuredata to generate a data set representative of particle motion atpositions between the streamers (i.e., interpolation) and substantiallyfree of surface ghosts (i.e., deghosting). This method differs from theprevious method in the sense that the step of denoising in the p modelis skipped, and the output data set represents particle motion insteadof pressure data.

Note that there can be general benefit of processing particle motiondata rather than pressure data because particle motion data providesinformation about the orientation of the wave-field that can be usefulfor demultiple, interference noise removal, etc. Further, each particlemotion component (for example, vx, vy, vz) at different points in theprocessing sequence may be processed independently or jointly. Aftermigration, the data may be combined to simulate a pressure wave-field.

According to still another embodiment, it is possible to perform jointinterpolation and deghosting using multicomponent streamer measurements.The method includes a step of receiving particle motion data andpressure data as in the previous methods, and a step of processingparticle motion data and pressure data to generate a data setrepresentative of a pressure wave-field at positions between thestreamers (interpolation) and substantially free of up-going primaryenergy. The difference with the previous method is that instead ofparticle motion data, pressure data is calculated so that it is free ofprimaries and not of ghosts, i.e., the calculated pressure data includesghosts and not primaries. Note that working with down-going (ghost) datacan produce similar results to working with up-going (primary) data. Insome environments, better imaging can be achieved using mirror migrationof ghost energy rather than regular migration of primary energy. Theoutput data may be at the same datum as the input or a new datum, i.e.,combining removing up-going energy, redatuming, and interpolation.

According to yet another embodiment illustrated in FIG. 17, outputseismic data may be generated that includes vector rotation andobliquity correction using multicomponent streamer measurements. Morespecifically, as illustrated in the figure, in step 1700, particlemotion data and pressure data is received as in the previousembodiments. In step 1702, particle motion data and pressure data areprocessed to perform interpolation of the pressure wave-field and/orvector rotation with or without obliquity correction (see equation (6)and associated description and also FIG. 8 and associated description).The processing step determines a model p based on input hydrophone andparticle velocity data d as before, e.g., tau-p_(x)-p_(y) model.However, instead of using this model p for combined interpolation anddeghosting, it may be used only for interpolation/regularization ofpressure and particle velocity data. Particle velocity data may be at adifferent orientation relative to input data and, thus, the particlevelocity data may or may not be compensated for obliquity correction.

In one application, output data may not be interpolated. In this case,particle motion data may or may not be corrected for obliquity. Theoutput V_(z) measurements may subsequently be summed with pressure datain the data space d to perform wave-field separation through P-V_(z)summation. Wavefield separation through P-V_(z) summation is known inthe field and, thus, this process is omitted herein.

Alternatively, the summation may be made post- or during migration. Instill another application, pressure P and particle velocity V_(z) datamay be processed through migration. Four migrations may be used:migration and mirror migration for both P and Vz data. Following thesesteps, a (quadruple) joint deconvolution or modeling approach could beused to estimate the reflectivity (ghost-free data). The jointdeconvolution or modeling approach generates a single reflectivity modelthat simultaneously satisfies all four datasets.

According to still another embodiment, collected data may be processedto obtain only deghosted data, i.e., no interpolation step. According tothis method, there is a step of receiving particle motion data andpressure data as in the previous methods, and a step of processingparticle motion data and pressure data to generate a data setrepresentative of a pressure wave-field at positions along thestreamers, and the pressure wave-field is substantially free of surfaceghosts. The output data set may include traces at the positions of thereceivers along streamers or between the receivers, or both.

According to yet another embodiment, there is a method for eliminationof cross-talk noise (if a simultaneous shooting scheme is used) orinterference noise from another survey. The method includes a step ofreceiving particle motion data and pressure data as discussed withregard to previous methods, a step of processing particle motion dataand pressure data to remove simultaneous shooting cross-talk noise orinterference noise, and a step of outputting pressure and particlevelocity data along the streamers, free of ghosts. The processing stepmay take place in the model domain and may uses traditional algorithmsfor identifying cross-talk noise or interference noise.

According to yet another embodiment, it is possible to have a methodthat receives the same data as in previous methods, then processesparticle motion data and pressure data to generate a data setrepresentative of a pressure wave-field at a new first datum, and thenspatially resamples the data indicative of particle motion and pressuredata at a second new datum. Redatum data may include surface ghosts, orit may be substantially free of surface ghosts, or it may besubstantially free of up-going (primary) energy.

According to another embodiment, there is a method that combinesdeghosting with spatial resampling. For example, the method includes astep of receiving particle motion data and pressure data as in theprevious methods and a step of processing particle motion data andpressure data to generate a data set representative of a pressurewave-field substantially free of surface ghosts at new spatialpositions. New spatial positions may be at a new depth, or at a newdepth and resampled in lateral position.

According to yet another embodiment, there is a method for processinginput seismic data d that includes a step of receiving input seismicdata d recorded in a first domain by seismic receivers that travel inwater, with the input seismic data d including primary and surface ghostwave-fields, wherein the input seismic data includes both pressure andparticle motion measurements; a step of generating a model p in a seconddomain to describe input seismic data d; and a step of processing modelp to obtain multiples in the second domain, wherein the multiples arerepresentative of primary wave-fields reflected at the water freesurface or in the earth (internal multiples). The method may furtherinclude transforming the multiples back to the first domain, andsubtracting the multiples from input seismic data d to obtain final datad′ based on which a final image of the earth is generated. In oneapplication, the step of processing involves applying a convolution inthe second domain to estimate multiples energy.

The above-discussed procedures and methods may be implemented in acomputing device as illustrated in FIG. 18. Hardware, firmware, softwareor a combination thereof may be used to perform the various steps andoperations described herein. Computing device 1800 of FIG. 18 is anexemplary computing structure that may be used in connection with such asystem.

Exemplary computing device 1800 suitable for performing the activitiesdescribed in the exemplary embodiments may include a server 1801. Such aserver 1801 may include a central processor (CPU) 1802 coupled to arandom access memory (RAM) 1804 and to a read-only memory (ROM) 1806.ROM 1806 may also be other types of storage media to store programs,such as programmable ROM (PROM), erasable PROM (EPROM), etc. Processor1802 may communicate with other internal and external components throughinput/output (I/O) circuitry 1808 and bussing 1810 to provide controlsignals and the like. Processor 1802 carries out a variety of functionsas are known in the art, as dictated by software and/or firmwareinstructions.

Server 1801 may also include one or more data storage devices, includinghard drives 1812, CD-ROM drives 1814 and other hardware capable ofreading and/or storing information, such as DVD, etc. In one embodiment,software for carrying out the above-discussed steps may be stored anddistributed on a CD-ROM or DVD 1816, a USB storage device 1818 or otherform of media capable of portably storing information. These storagemedia may be inserted into, and read by, devices such as CD-ROM drive1814, disk drive 1812, etc. Server 1801 may be coupled to a display1820, which may be any type of known display or presentation screen,such as LCD, plasma display, cathode ray tube (CRT), etc. A user inputinterface 1822 is provided, including one or more user interfacemechanisms such as a mouse, keyboard, microphone, touchpad, touchscreen, voice-recognition system, etc.

Server 1801 may be coupled to other devices, such as sources, detectors,etc. The server may be part of a larger network configuration as in aglobal area network (GAN) such as the Internet 1828, which allowsultimate connection to various landline and/or mobile computing devices.

The above embodiments have presented various algorithms for processinginput seismic data d. Those embodiments are now summarized for a betterunderstanding of the claimed methods. Literal references are providedfor each embodiment and numeral references are provided for the variousfeatures associated with a given embodiment. The following embodimentsare just exemplary and not intended to limit the invention. The featuresfor the embodiments are listed with a corresponding numeral referenceand each feature may work with any other feature of a respectiveembodiment. Note that all these features are disclosed above and thefollowing section only organizes these features in an easy to followway. All the features listed next may be implemented into a computingdevice such that these calculations are automatically performed. Thus,the processor of a computing device may be configured to execute any ofthe following features, in combination or not. However, the followinglist of features is not intended to be exhaustive and other combinationsof these features are contemplated.

Embodiment A

1. According to this embodiment, also illustrated in FIG. 19, there is amethod for processing input seismic data d, and includes a step 1900 ofreceiving the input seismic data d recorded in a first domain by seismicreceivers that travel in water, the input seismic data d includingup-going and down-going wave-fields; a step 1902 of generating a model pin a second domain to describe the input seismic data d; and a step 1904of processing the model p to obtain an output seismic dataset indicativeof the down-going wave-field and substantially free of the up-goingwave-field.

2. The input seismic data d includes only pressure measurements.

3. The input seismic data d includes only particle motion measurements.

4. The input seismic data d includes both pressure and particle motionmeasurements.

5. The output seismic dataset is indicative of pressure measurementsand/or particle motion measurements.

6. The first domain is a time-space domain.

7. The second domain is one of a radon domain, frequency-wave numberdomain, tau-p domain, parabolic domain, shifted hyperbola domain,singular value decomposition, rank reduction and curvelet domain.

8. The step of processing the model p comprises:

applying an L′ transform to the model p to obtain the output seismicdataset.

9. The step of processing also includes removing random, coherent orimpulsive noise.

10. The coherent noise is cross-talk noise, interference noise ormultiple energy.

11. The impulsive noise is cross-talk noise or interference noise.

12. An amount of noise is reduced by scaling or filtering energyassociated with the model p prior to or during the application of the L′transform.

13. The L′ transform also includes a receiver rotation or obliquitycorrection.

14. The output seismic dataset indicative of the down-going wave-fieldis generated at input positions.

15. The seismic dataset indicative of the down-going wave-field issubtracted from the input seismic data d to obtain data d′ to be usedfor generating a final image of a surveyed subsurface.

16. The output seismic dataset indicative of the down-going wave-fieldis generated at positions different from the input seismic data.

17. The positions are in-between the receivers.

18. The positions are at different depths than the input seismic data.

19. The different depths are at the sea surface.

20. The positions are selected to match positions of receivers fromanother seismic survey.

21. The positions are equidistant from streamers on which the receiversare distributed.

22. The positions are distributed on a regular grid.

23. The seismic dataset indicative of the down-going wave-field isdirectly used for generating a final image of a surveyed subsurface.

24. The step of processing the model p comprises:

generating a seismic dataset indicative of the up-going wave-field andsubtracting the up-going data from the input data to generate an outputdataset substantially free of up-going energy.

Embodiment B

1. According to this embodiment, also illustrated in FIG. 20, there is amethod for processing input seismic data d, the method comprising a step2000 of receiving the input seismic data d recorded, at a first datum,in a first domain, by seismic receivers that travel in water, the inputseismic data d including up-going and down-going wave-fields, a step2002 of generating a model p in a second domain to describe the inputseismic data d, and a step 2004 of processing the model p to obtain anoutput seismic dataset at a second datum, different from the firstdatum.

2. The output seismic data is substantially free of the down-goingwave-field.

3. The output seismic data is substantially free of the up-goingwave-field.

4. The output seismic data contains both the up-going wave-field anddown-going wave-field.

5. The input seismic data d includes particle motion measurements.

6. The input seismic data d includes pressure and particle motionmeasurements.

7. The first datum is constant.

8. The first datum varies along an inline direction.

9. The first datum slants along an inline direction.

10. The first datum varies along an inline direction based on a curvedshape.

11. The curved shape is a sinusoidal shape.

12. The first datum varies along a cross-line direction.

13. The second datum is constant.

14. The second datum varies along an inline direction.

15. The second datum slants along an inline direction.

16. The second datum varies along an inline direction based on a curvedshape.

17. The curved shape is a sinusoidal shape.

18. The second datum varies along a cross-line direction.

19. The second datum is at the water surface level.

20. Plural output seismic datasets are generated at plural receiverdepths.

21. The plural receiver depths are selected to maximize spatialresolution.

22. The output seismic dataset is generated at positions different frompositions of the input seismic data.

23. The output positions are selected to match positions of receiversfrom another seismic survey.

24. The output positions are equidistant from streamers on which thereceivers are distributed.

25. The output positions are distributed on a regular grid.

26. The output seismic dataset is used for generating a final image of asurveyed subsurface.

27. The first domain is a time-space domain.

28. The second domain is one of a radon domain, frequency-wave numberdomain, tau-p domain, parabolic domain, singular value decompositiondomain, rank reduction domain, shifted hyperbola domain and curveletdomain.

29. The step of processing the model p comprises:

applying an L′ transform to the model p to obtain a seismic dataset atthe second datum.

30. An amount of noise is reduced by filtering or scaling energyassociated with the p model before or when the L′ transform is applied.

31. The L′ transform also includes resampling.

32. The output seismic dataset is indicative of pressure measurements orparticle motion measurements.

33. The L′ transform also includes a receiver rotation or obliquitycorrection.

Embodiment C

1. According to this embodiment, also illustrated in FIG. 21, there is amethod for processing input seismic data d, the method including a step2100 of receiving the input seismic data d recorded, in a first domain,by seismic receivers that travel in water, a step 2102 of generating amodel p in a second domain to describe the input seismic data d, and astep 2104 of processing the model p to generate an output particlemotion dataset.

2. The input seismic data d includes only pressure measurements.

3. The input seismic data d includes only particle motion measurements.

4. The input seismic data d includes both pressure and particle motionmeasurements.

5. The output particle motion dataset is not corrected for obliquity.

6. The output particle motion dataset is corrected for obliquity.

7. The obliquity is defined by an angle between a respective wave-fieldpropagation direction and a receiver orientation.

8. The receiver orientation is defined by an angle relative to gravity.

9. The receiver orientation is defined by an angle relative to thenominal shooting direction.

10. The output particle motion data is re-orientated.

11. The processing step comprises:

wave-field reconstruction of the pressure wave-fields based on the modelp.

12. The pressure wave-fields are reconstructed at the same positions asthe input seismic data, d.

13. Incoming wave-fields are reconstructed at new receiver positions.

14. The new positions are at different depths to the input data.

15. The new positions are in-between the streamers.

16. The output particle motion dataset includes both an up-goingwave-field and a down-going wave-field.

17. The output particle motion dataset is substantially free of thedown-going wave-field.

18. The output particle motion dataset is substantially free of theup-going wave-field.

19. The first domain is time-offset/x-offset/y and the second domain istau-slowness/x-slowness/y, where offset/x is a distance between a sourcegenerating the input seismic data d and a corresponding receiver along afirst direction and offset/y is a distance between the source and thecorresponding receiver along a second direction, which is substantiallyperpendicular on the first direction.

20. The output particle motion dataset is combined with pressure data inthe first domain to obtain wave-field separation.

21. The output particle motion dataset is summed with hydrophone data inthe first domain to obtain wave-field separation.

22. The output particle motion dataset is subtracted from hydrophonedata in the first domain to obtain wave-field separation.

23. The first domain is a time-space domain and the second domain is oneof a radon domain, frequency-wave number domain, tau-p domain, parabolicdomain, hyperbolic domain, singular value decomposition domain, rankreduction domain and curvelet domain.

24. The step of generating a model p comprises:

computing the model p by solving an inverse problem based on an Ltransform; and

applying an L′ transform to the model p to obtain the output particlemotion dataset, wherein the L′ transform combines the obliquity andwave-field reconstruction.

25. The step of processing the model includes removing noise.

26. The step of removing noise includes removing coherent, impulsive orrandom noise.

27. The noise relates to multiple, cross-talk, or interference noise.

28. An amount of noise is reduced by scaling/filtering the model p orcontrolling weights when the L′ transform is applied.

29. The output particle motion dataset is used for generating a finalimage of a surveyed subsurface.

30. The output particle motion dataset is subtracted from recordedparticle motion data and the remaining energy is denoised.

Embodiment D

1. According to this embodiment, also illustrated in FIG. 22, there is amethod for processing input seismic data d, the method including a step2200 of receiving the input seismic data d recorded, in a first domain,by seismic receivers that travel in water, the input seismic data dincluding pressure and particle motion measurements, a step 2202 ofgenerating a model p in a second domain to describe the input seismicdata d; and a step 2204 of processing the model p to generate an outputseismic dataset with attenuated noise.

2. The noise is coherent, impulsive or random.

3. The noise relates to cross-talk noise, interference noise or multiplenoise.

4. The cross-talk noise relates to energy coming from a second vesselinvolved in the same seismic acquisition.

5. The interference noise relates to a vessel not involved in the sameseismic acquisition.

6. The step of processing comprises:

removing noise in the model p by scaling.

7. The method further comprises:

removing the noise based on the non-coherent nature of the model p inthe second domain.

8. An amount of noise is reduced by controlling sparseness weights whenan L transform is applied.

9. The first domain is a time-space domain.

10. The second domain is one of a radon domain, frequency-wave numberdomain, tau-p domain, parabolic domain, hyperbolic domain, singularvalue decomposition domain, rank reduction domain, slowness-shotpointdomain and curvelet domain.

11. The step of generating a model p comprises:

computing the model p by solving an inverse problem based on an Ltransform; and

applying an L′ transform to the model p to obtain the output seismicdataset with attenuated noise.

12. The L′ transform applies model masking or scaling or filtering toenergy associated with the model p.

13. The L′ transform also includes receiver rotation or obliquitycorrection.

14. The output seismic dataset is at the same positions as the inputdata.

15. The output seismic dataset is at different positions to input data.

16. The output seismic dataset is obtained after applying a step ofwavefield separation.

17. The output seismic dataset contains up-going and down-goingwavefields.

18. The output seismic dataset is indicative of a pressure measurement.

19. The output seismic dataset is indicative of a particle motionmeasurement.

20. The output seismic dataset with attenuated noise is used forgenerating a final image of a surveyed subsurface.

Embodiment E

1. According to this embodiment, also illustrated in FIG. 23, there is amethod for processing input seismic data d, the method including a step2300 of receiving the input seismic data d recorded by seismic receiversthat travel in water, the input seismic data d including particle motionmeasurements, a step 2302 of receiving receiver orientation data that isindicative of seismic receiver orientations, a step 2304 of associatingthe particle motion measurements with the receiver orientation data, anda step 2306 of generating an output seismic data based on theassociation of the particle motion measurements with the receiverorientation data. The seismic receiver orientations vary in time and theoutput seismic data includes a wavefield reconstruction of the inputdataset.

2. The input seismic data further includes pressure measurements.

3. The output seismic data is related to pressure or particle motionmeasurements.

4. The wavefield reconstruction generates up-going wave-fields.

5. The wavefield reconstruction generates down-going wave-fields.

6. The wavefield reconstruction generates up-going and down-goingwave-fields.

7. The wavefield reconstruction of the input data is at the samepositions as the input data.

8. The wavefield reconstruction reconstructs the wave-fields at desiredpositions that are different from positions relating to input seismicdata d.

9. The desired positions are at different depths to the input receivers.

10. The desired positions are in-between the receivers.

11. The desired positions are selected to match positions of receiversfrom another seismic survey.

12. The desired positions are equidistant from streamers on which thereceivers are distributed.

13. The desired positions are distributed on a regular grid.

14. The seismic receiver orientations vary from receiver to receiveralong a streamer and the receiver orientation data includes orientationinformation about each receiver.

15. The seismic receiver orientations vary from a group of receivers toanother group of receivers along a streamer and the receiver orientationdata includes orientation information about the group of receivers.

16. Receiver data and the receiver orientation data has beenpre-processed with an initial receiver orientation rotation.

17. The receiver orientation data is constant for each trace.

18. The receiver orientation data varies during the recording of eachtrace.

19. The seismic receiver orientations are distributed along more than avertical direction and a cross-line direction.

20. The seismic receiver orientations are substantially perpendicular tothe streamer.

21. The input seismic data d is recorded in a time-space domain.

22. The method further comprises:

generating a model p to describe the input seismic data d; and

applying an L′ transform to the model p to obtain the output seismicdata.

23. The model p is generated in one of a radon domain, frequency-wavenumber domain, tau-p domain, parabolic domain, shifted hyperbola domain,singular value decomposition, rank reduction and curvelet domain.

24. The step of applying an L′ transform also includes removing random,coherent or impulsive noise.

25. The coherent noise is cross-talk noise, interference noise ormultiple energy.

26. The impulsive noise is cross-talk noise or interference noise.

27. An amount of noise is reduced by scaling or filtering energyassociated with the model p prior to or during the application of the L′transform.

28. The L′ transform also includes a receiver rotation or obliquitycorrection.

29. The output seismic data is subtracted from the input seismic data dto obtain data d′ to be used for generating an image of a surveyedsubsurface.

30. The output seismic data is directly used for generating an image ofa surveyed subsurface.

Embodiment F

1. According to this embodiment, also illustrated in FIG. 24, there is amethod for processing input seismic data d, the method including a step2400 of receiving the input seismic data d recorded in a first domain byseismic receivers that travel in water, wherein the input seismic dataincludes both pressure and particle motion measurements, a step 2402 ofgenerating a model p in a second domain to describe the input seismicdata d, and a step 2404 of processing the model p to separate multiplesand primaries in the second domain, wherein the multiples is multipleenergy reflected at the free surface or rock interface layers in thesubsurface.

2. The step of processing involves scaling down energy which is notmultiple energy.

3. The step of processing involves scaling down energy which is multipleenergy.

4. The step of processing involves applying deconvolution in the seconddomain to separate primary and multiples energy.

5. The step of processing involves applying a convolution in the seconddomain to estimate multiples energy.

6. The method further comprises:

transforming the multiples back to the first domain; and

subtracting the multiples from the input seismic data d to obtain finaldata d′ based on which a final image of the earth is generated.

7. The method further comprises:

subtracting the multiples from data in the second domain; and

applying an operator L′ to transform the data free of multiples from thesecond domain back to the first domain.

8. The first domain is a time-space domain.

9. The second domain is one of a radon domain, frequency-wave numberdomain, tau-p domain, parabolic domain, hyperbolic domain, singularvalue decomposition domain, rank reduction domain and curvelet domain.

10. The step of generating a model p comprises:

computing the model p by solving an inverse problem based on an Ltransform;

separating the primaries and the multiples in the second domain; and

applying an L′ transform to the model p to obtain the multiples or theprimaries in the first domain.

11. The L′ transform includes wavefield separation.

12. The multiples or the primaries are generated at the same positionsas the input data.

13. The multiples or the primaries are generated at positions differentto the input data.

14. The multiples or the primaries are generated at positions to matchpositions of receivers from another seismic survey.

15. The positions are equidistant from input streamers on which thereceivers are distributed.

16. The positions are distributed on a regular grid.

17. The positions are at different depths to the input data.

18. The positions are in-between the streamers.

19. The output data is indicative of pressure measurements.

20. The output data is indicative of particle motion measurements.

21. The output data is corrected for obliquity.

22. The output data is re-oriented.

Embodiment G

1. According to this embodiment, also illustrated in FIG. 25, there is amethod for processing input seismic data d, the method including a step2500 of receiving the input seismic data d recorded in a first domain byseismic receivers that travel in water, the input seismic data dincluding pressure and particle motion measurements, a step 2502 ofgenerating a model p in a second domain to describe the input seismicdata d, wherein the model p is obtained by solving an inverse problembased on an L transform, and a step 2504 of processing with amathematical transform L′ the model p to obtain, in the first domain, anoutput seismic data having a characteristic imparted by the transformL′. The mathematical transform L′ is different from the mathematicaltransform L.

2. The input seismic data includes up-going and down-going wave-fields.

3. The input seismic data includes only an up-going wave-field.

4. The input seismic data includes only a down-going wave-field.

5. The output seismic dataset is indicative of pressure measurements.

6. The output seismic dataset is indicative of particle motionmeasurements with or without obliquity correction.

7. The characteristic is related to pressure wave-fields and/or particlemotion wave-fields being interpolated among receiver positions and beingsubstantially free of the up-going wave-fields.

8. The characteristic is related to pressure wave-fields and/or particlemotion wave-fields substantially free of the down-going wave-field andinterpolated among receiver positions.

9. The characteristic is related to pressure wave-fields and/or particlemotion wave-fields being interpolated along streamers including theseismic receivers.

10. The interpolated data is indicative of hydrophone and particlemotion data is summed to perform wavefield separation.

11. The characteristic is related to pressure wave-fields and/orparticle motion wave-fields being calculated at a new datum relative toa datum of the seismic receivers.

12. Pressure wave-fields and/or particle motion wave-fields arespatially resampled at another new datum.

13. The characteristic is related to pressure wave-fields and/orparticle motion wave-fields being wave-field separated and calculated ata new datum relative to a datum of the seismic receivers.

14. The method further comprises:

denoising the model p prior to applying the L′ transform.

15. The characteristic is related to pressure wave-fields and/orparticle motion wave-fields being interpolated at positions in-betweenthe input seismic receivers.

16. The characteristic is related to pressure wave-fields and/orparticle motion wave-fields being interpolated at positions selected tomatch positions of receivers from another seismic survey.

17. The positions are equidistant from input streamers on which thereceivers are distributed.

18. The positions are distributed on a regular grid.

19. The first domain is a time-space domain and the second domain is oneof a radon domain, frequency-wave number domain, tau-p domain, parabolicdomain, hyperbolic domain, singular value decomposition domain, rankreduction domain and curvelet domain.

20. The seismic dataset indicative of up-going and down-goingwave-fields is used for generating a final image of a surveyedsubsurface.

Embodiment H

1. According to this embodiment, also illustrated in FIG. 26, there is amethod for processing input seismic data d, the method including a step2600 of receiving the input seismic data d recorded in a first domain byseismic receivers that travel in water, a step 2602 of generating amodel p in a second domain, at a datum different from the input seismicdata d, to describe the input seismic data d, and a step 2604 ofprocessing the model p to obtain an output seismic dataset indicative ofa pressure wave-field.

2. The input seismic data d includes only pressure measurements orpressure and particle motion measurements.

3. The output seismic dataset is indicative of a pressure measurement ora particle motion measurement.

4. The first domain is a time-space domain.

5. The second domain is one of a radon domain, frequency-wave numberdomain, tau-p domain, parabolic domain, shifted hyperbola domain,singular value decomposition domain, rank reduction domain and curveletdomain.

6. The step of processing the model p comprises:

applying an L′ transform to the model p to obtain the output seismicdataset.

7. The L′ transform does not include vector rotation or obliquitycorrection.

8. The L′ transform also includes vector rotation or obliquitycorrection.

9. The step of processing the model p comprises:

applying an L′ transform to the model p to obtain an output datasetcontaining up-going and down-going energy.

10. The step of processing the model p comprises:

applying an L′ transform to the model p to obtain an output datasetsubstantially free of down-going energy.

11. The step of processing the model p comprises:

applying an L′ transform to the model p to obtain an output datasetsubstantially free of up-going energy.

12. The step of processing also includes removing random, coherent orimpulsive noise.

13. The coherent noise is cross-talk noise, interference noise ormultiple energy.

14. The impulsive noise is cross-talk noise or interference noise.

15. An amount of noise is reduced by scaling or filtering prior to orduring the application of the L′ transform.

16. The output seismic dataset indicative of the pressure wave-field isgenerated at the same positions as the input seismic data.

17. The output seismic dataset indicative of the pressure wave-field isgenerated at positions different from the input seismic data.

18. The positions are at different depths than the input seismic data.

19. The positions are designed to match positions of receivers fromanother seismic survey.

20. The positions are equidistant from input streamers on which thereceivers are distributed.

21. The positions are on a regular grid.

22. The output seismic dataset indicative of the pressure wave-field issubtracted from the input seismic data d to obtain data d′ to be usedfor generating a final image of a surveyed subsurface.

23. The output seismic dataset indicative of the pressure wave-field isadded to the input seismic data d to obtain data d′ to be used forgenerating a final image of a surveyed subsurface.

24. The seismic dataset indicative of the pressure wave-field isdirectly used for generating a final image of a surveyed subsurface.

25. The step of generating a model is based on an L transform whichincorporates a differentiation in time or space.

26. The step of generating a model is based on an L transform whichincludes a frequency filter term to account for different signal tonoise ratios for pressure and particle motion data.

27. Measurements of the particle motion receivers are substantiallyorientated vertically with gravity.

The disclosed exemplary embodiments provide a computing device, softwareinstructions and a method for seismic data processing. It should beunderstood that this description is not intended to limit the invention.On the contrary, the exemplary embodiments are intended to coveralternatives, modifications and equivalents, which are included in thespirit and scope of the invention as defined by the appended claims.Further, in the detailed description of the exemplary embodiments,numerous specific details are set forth in order to provide acomprehensive understanding of the claimed invention. However, oneskilled in the art would understand that various embodiments may bepracticed without such specific details.

Although the features and elements of the present exemplary embodimentsare described in the embodiments in particular combinations, eachfeature or element can be used alone without the other features andelements of the embodiments or in various combinations with or withoutother features and elements disclosed herein.

This written description uses examples of the subject matter disclosedto enable any person skilled in the art to practice the same, includingmaking and using any devices or systems and performing any incorporatedmethods. The patentable scope of the subject matter is defined by theclaims, and may include other examples that occur to those skilled inthe art. Such other examples are intended to be within the scope of theclaims.

What is claimed is:
 1. A method for processing input seismic data dassociated with a surveyed subsurface, the method comprising: receivingthe input seismic data d recorded in a first domain by seismic receiversthat travel in water, the input seismic data d including pressure andparticle motion measurements; generating a model p in a second domain bysolving an inverse problem for the input seismic data d, whereinapplying an L transform to the model p describes the input data d;applying an L′ transform, which is different from the L transform, tothe model p to obtain an output seismic data in the first domain, theoutput seismic data having a characteristic imparted by the transformL′; and generating an image of the surveyed subsurface based on theoutput seismic dataset, wherein the input seismic data includes up-goingand down-going wave-fields, and the characteristic is related topressure wave-fields and/or particle motion wave-fields interpolated atpositions in-between the input seismic receivers.
 2. The method of claim1, wherein the pressure wave-fields and/or particle motion wave-fieldsare interpolated along streamers carrying the seismic receivers.
 3. Themethod of claim 1, wherein the output seismic data is indicative ofhydrophone and particle motion data summed for up-going and down-goingwave-field separation.
 4. The method of claim 1, wherein the pressurewave-fields and/or particle motion wave-fields are substantially free ofup-going wave-field components.
 5. The method of claim 1, wherein theoutput seismic dataset is indicative of pressure measurements.
 6. Themethod of claim 1, wherein the output seismic dataset is indicative ofparticle motion measurements.
 7. The method of claim 1, wherein theoutput seismic dataset is indicative of particle motion measurementswith an obliquity correction.
 8. The method of claim 7, wherein theobliquity is defined by an angle between a detected seismic wave-fielddirection and a receiver orientation, the receiver orientation isdefined by an angle relative to gravity or the nominal shootingdirection.
 9. The method of claim 1, wherein the pressure wave-fieldsand/or particle motion wave-fields are substantially free of down-goingwave-field components and interpolated among receiver positions.
 10. Themethod of claim 1, wherein the pressure wave-fields and/or particlemotion wave-fields are calculated at a new datum relative to a datum ofthe seismic receivers.
 11. The method of claim 10, wherein the pressurewave-fields and/or particle motion wave-fields are spatially resampledat another new datum.
 12. The method of claim 1 wherein, the pressurewave-fields and/or particle motion wave-fields are separated intoup-going and down-going components and calculated at a new datumdifferent from a datum of the seismic receivers.
 13. The method of claim1, further comprises: denoising the model p prior to applying the L′transform.
 14. The method of claim 1, wherein the pressure wave-fieldsand/or particle motion wave-fields are interpolated at positionsselected to match positions of receivers from another seismic survey.15. The method of claim 1, wherein the positions are equidistant fromstreamers on which the seismic receivers are distributed.
 16. The methodof claim 1, wherein the positions are distributed on a regular grid. 17.The method of claim 1, wherein the first domain is a time-space domainand the second domain is one of a radon domain, a frequency-wave numberdomain, a tau-p domain, a parabolic domain, a hyperbolic domain, asingular value decomposition domain, a rank reduction domain and acurvelet domain.
 18. The method of claim 1, wherein the output seismicdataset is indicative of up-going wave-field components only.
 19. Acomputing device for processing input seismic data d, the devicecomprising: an interface configured to receive the input seismic data drecorded, in a first domain, by seismic receivers that travel in waterthe input seismic data d including pressure and particle motionmeasurements; and a processor connected to the interface (1808), theprocessor being configured to, generate a model p in a second domain bysolving an inverse problem for the input seismic data d, whereinapplying an L transform to the model p describes the input data d; applyan L′ transform, which is different from the L transform, to the model pto obtain an output seismic data in the first domain, the output seismicdata having a characteristic imparted by the transform L′; and generatean image of the surveyed subsurface based on the output seismic dataset,wherein the input seismic data includes up-going and down-goingwave-fields, and the characteristic is related to pressure wave-fieldsand/or particle motion wave-fields interpolated at positions in-betweenthe input seismic receivers.
 20. A non-transitory computer readablemedium including computer executable instructions, wherein theinstructions, when executed by a processor, implement a method forprocessing input seismic data d, the method comprising: receiving theinput seismic data d recorded in a first domain by seismic receiversthat travel in water, the input seismic data d including pressure andparticle motion measurements; generating a model p in a second domain bysolving an inverse problem for the input seismic data d, whereinapplying an L transform to the model p describes the input data d;applying an L′ transform, which is different from the L transform, tothe model p to obtain an output seismic data in the first domain, theoutput seismic data having a characteristic imparted by the transformL′; and generating an image of the surveyed subsurface based on theoutput seismic dataset, wherein the input seismic data includes up-goingand down-going wave-fields, and the characteristic is related topressure wave-fields and/or particle motion wave-fields interpolated atpositions in-between the input seismic receivers.